The/My father who

I won't find supporting references for this, but I'd label

• The father who had planned my life to the point of my unsought arrival in Brighton took it for granted that in the last three weeks of his legal guardianship I would still act as he directed..

a literary usage where a defining structure (cf 'The man who killed Liberty Valance') is used to vastly strengthen the rather weaker 'adjunct' 'My father, who had planned my life ...' implying that this is far from being tagged-on additional information. So syntactically classifiable as 'defining', but semantically not identifying a particular father(!) but emphasising the necessity of the clause, showing it's far from being '..., Oh, and by the way, he had planned my life ...').

Using 'my' instead of 'the' needs a comma on each side of the now non-defining (both syntactically and semantically) relative clause. Perfectly grammatical, but far weaker.

The question seems to be whether 'my' is allowed as determiner in an integrated relative clause - specifically of the type mentioned by H&P in their Student's Introduction to English Grammar where the relative clause is not restrictive: where it does not express a distinguishing property that helps us pick out the referent of the NP from a potential pool of referents.

The father who had planned my life to the point of my unsought arrival in Brighton took it for granted that in the last three weeks of his legal guardianship I would still act as he directed.

The defining characteristic of this case seems to be that the relative clause does not help in pointing out the referent of the NP from among a group of potential candidates, as this has already been done by context - in the above example, exempting the case where two fathers existed, it is abundantly clear that the relative clause does not help us to figure out which father is being referred to.

H&P give three main distinctions between integrated and supplementary clauses (CaGEL p1058) which I've summarized here (they go into much more detail and this is not meant as a substitute for the full analysis, just a signpost):

a) Prosody and punctuation

slight pause/change of intonation, commas

b) Syntax

supplementary only allow wh and that (marginal)

c) Meaning

integrated relative clauses have meaning integral to the clause containing it

They later go on to analyze the given relative from each perspective on p1065 (I've taken the liberty of reformatting):

a) it cannot be omitted or spoken on a separate intonation contour

b) allows that as an alternant of who (albeit somewhat less favoured)

c) does not serve to distinguish this father from other fathers of the narrator: he has only one father. The reason for presenting the content of the relative clause as an integral part of the message is not, therefore, that it expresses a distinguishing property but that it explains why the father took it for granted that the son would do as he was told

The case with an integrated non-restrictive relative has plenty of examples amenable to the same analysis:

The clause which read "the boy known as Roger Lennox Irving" did not especially trouble him now, though he could not then forgive the father who had wronged him so (Roger Irving's Ward; Holmes, Mary Jane; 1871)

Why has anything been hidden from me -- the father who loves her better than his life? (Elsie's Vacation and After Events; Finley, Martha; 1891)

It had been even so, and Arthur Ferris left his girl wife, still a stranger to him, in the care of the father who demanded the New York deal with the senatorial ally as the price of the strangely deferred honeymoon joys. (The Midnight Passenger; Savage, Richard; 1900)

a child came running with outstretched arms and piteous voice -- a frightened child, weeping for the father who had thrown himself headlong into peril to save another's life and who, perhaps, had lost his own (The Littlest Rebel; Peple, Edward Henry; 1914)

The father who had once heartened him as a child now lay helpless and pathetic on the bed before him.(He gave him a stone; Charles W. (Charles Wright) Ferguson; 1938)

For my sake, for the sake of the father who took the wrong turns, take the right ones, and carry my blessing and my justification with you.(Caine Mutiny; Herman Wouk; 1952)

He wipes his mouth with a handkerchief, and gets his breath back. Pause. Bertram speaks to the father who can not hear him.) p. 53 BERTRAM I was your boy, not your dog. What I wanted, more than anything else, you gave to Captain Bob. Oh, this life. (The Captivity of Pixie Shedman; Linney, Romulus; 1981)

The in the examples above seems readily replaceable with genitive pronouns her, his, my, etc. If it does not seem so at first, compare with the following which also exemplify this sort of integrated relative.

My father who was far from being abundantly supplied with any other possession except a numerous family of sons and daughters, could not afford to expend much worldly substance upon my education. (O'Halloran; McHenry, James; 1824)

Belfort remained in England under an assumed name, and in disguise visited his father who was a baronet and the son of an earl. (The Spanish Galleon; Ingraham, J. H.; 1844)

My father who had received in this scene a great shock, began to fail so rapidly, he demanded my constant care (A Strange Disappearance; Green, Anna Katharine; 1880)

With the exception of his father who had infinite faith in the shrewdness and ability of his son, the men he wanted to impress were only amused. (Poor White; Anderson, Sherwood; 1920)

he still held fast to the determination to go, and to find fortune somewhere along the trail of his own making; and to ask help from no man, least of all his father who had told him to go. (Cow-Country; Bower, B. M.; 1921)

He knew, like herself, that she was martyred, and he hated his father who lived in the same house, cold and aloof, untouched by the emotional tempests which were forever sweeping over him. (Tropic of Cancer; Miller, Henry; 1934)

From now on Professor Tucker will be the one to speak officially for our new school. All I can say is I shall work hard to prove worthy of my father who introduced the bill to make this school possible. (Whatever the Battle Be; Edmonds, Randolph; play script, 1950)

You didn't work your way into the business. You were kicked into it by my father who gave you $25,000 when we got married. (Bewitched, 1965)

Hey, my father who coached football at Texas Lutheran for six years was strict on me when I was growing up. (Purple Pros: The Same Old Story; Sports Illustrated: November 16, 1981)

I am very sorry I should have the misfortune of being concerned in the affair, but however shall be more cautious for the future -- I will trust no man from henceforward -- no, not my father who begat me -- nor the brother who lay with me in my mother's womb (Finding a character's voice in Smollett's Roderick Random; Goode, Okey; Style, 1990 Fall)

I was in the hospital with double pneumonia in an intensive care unit. The whole thing just faded, disappeared. Whether I went through a tunnel, I'm not sure, but I found myself in a garden, which was emanating light as though it were in the earth all around. Everything in the place was magnificent. There, I met my father who had died many years before.(Near Death; CNN, spoken, 1992)

You like to fly, right? - I am terrified of flying. - Oh, perfect. Now you are in a two-seater and it is being flown by my father who weighs 462 pounds, because he lost 10. And he can't see out of his right eye or his left.(Next Stop Wonderland; movie, 1998)

When was it that I first took notice of it? I suppose it was some years ago when I suddenly awoke in my bed and squeezed my eyes shut in front of my father who had come to wake me. (Freedom; Janko Polic Kamov; Partisan Review, 1998 winter)

If I die, I get to see the two most important people in my life -- my father who died in 1987 and Andrew. (Richard Justice, Washington Post Staff Writer; Washington Post, 1998-11-26)

Without this accidental meeting, there would have been no letter or telephone call, no genuine consideration of my father who loved his brother and who was blameless in this situation, but not in all others. (From American Genius; Lynne Tillman; Literary Review, 2005 Spring)

His voice cracked on the word gift, as if he didn't deserve such gratitude, my father who will do just about anything for anyone, driving my mother crazy with all the favors he does for everyone else (including, as she likes to say, any random citizen of Outer Slobovia and its most godforsaken suburbs). (I see you everywhere; Glass, Julia; New York: Pantheon Books,Edition, 2008)

I wanted to wash her away, the smell, the memory, the thing that had happened but couldn't be, and I tried to climb the stairs, but I was too weak to stand, too light in the head, and I was afraid of the water, my father there, dead of a heart attack at fifty-seven, Leonard Lok crumpled in the shower, alone, two hours - my father who might have survived if Mother had been home, if Mother had heard a cry, if he hadn't hit his head so hard on the tile. (Tu B'Shvat: for the Drowned and the Saved; Thon, Melanie Rae; The Antioch Review, 2011)

Where was I to start, and go for advice? Naturally, I decided to consult my father who was an oncologic surgeon himself with a long experience treating solid tumors. （The seven attributes of the academic surgeon； Rosengart, Todd K; The American Journal of Surgery, 2017)

Given the above, it seems my is an acceptable alternative to the in the example from A Student's Introduction to English Grammar.

The one caveat appears to be that when a genitive pronoun is used instead of the, the acceptability of a that-relative drops significantly - apparently signaling a move in the direction of a supplementary. Still, the number of examples with no commas marking off an integrated wh relative in an NP with a genitive pronoun as determiner seems to be sufficient to support this use.

The father who had planned my life to the point of my unsought arrival in Brighton took it for granted that ...

(i) In the above (the the version) "who had planned my life to the point of my unsought arrival in Brighton" is a defining relative clause.

The NP subject is "The father who had planned my life to the point of my unsought arrival in Brighton"

(ii) In the "my" version ", who had planned my life to the point of my unsought arrival in Brighton," it is a descriptive, or non-defining clause. (Note the commas)

The NP subject is "My father"

The effect of the defining clause (i) is to separate the types of father that the writer's father represents by specifying one of the features of the father that separate this aspect of his character from others.

Consider: "I had last seen John when he and I were 13 years old. He had been fat and shy. We were now 43 years old and the John who I had known then was not the John who I saw now - he was slim, confident and obviously rich."

My has an adjectival effect and also separates this father from all other fathers: he is the father of the writer. However, ", who had planned my life to the point of my unsought arrival in Brighton, is merely additional and extraneous information - it may be omitted.

If the "my" version were a defining clause,"My father who had planned my life to the point of my unsought arrival in Brighton", it would imply that the author had another, physical father who had not "planned his life, etc". (And this is quite impossible.)

Here, can you replace The with My like this?

No. The sentences mean different things.