Can the adverb "strongly" be use with the verb "try"? If not, is there a clear reason?

I don't think there is anything wrong with "try strongly" per se. It is just not the idiom that we use. In English, when I was a kid, we used to say "I did such and such by accident", most young people now say "I did such and such on accident". It is just a change in the idiom of the language.

If you said "try strongly" people would certainly understand what you meant, but it wouldn't seem quite right. I am reminded of a question someone else asked a while ago which I answered:

What is wrong in "Please don't pluck the flowers" and other phrases used in the Indian subcontinent?

How does a learner learn what is right and wrong? Unfortunately, I think the only way is by reading, listening and speaking a lot of English with native speakers.

in the context of sports:

Sculthorpe raised hopes of a fightback with a typically strong try but a double from Brian Carney, Wigan's exuberant Irish flyer, put the seal on their night.

The Guardian

Bit of a stretch, since the phrase is strong not strongly.

Strong makes me think of physical strength. "A lot of effort" is typically conveyed by "hard". As you "work hard", so you can "try hard".