What is a word for someone who is speaking in a way to gain sympathy from you?

They are speaking supplicatingly

= Beseechingly, imploringly; in supplication

Oxford Lexico

From which, also consider:

Beseechingly, from beseeching : expressing or marked by earnest pleading or entreaty

“The wretched young man arose, and with a last beseeching glance at us walked from the room.”

Merriam Webster

And imploringly

in a way that expresses urgent or piteous pleading, as for aid or mercy; beseechingly:

"No! Don't kill the spider!" gasped my daughter, looking imploringly into my eyes.

Dictionary.com

I would say wretchedly, lamentably or pitiably - with the latter evoking pity (which is close to sympathy).

However, I have not often, if ever, heard pitiably used, whereas wretchedly and lamentably are more common.

Personally I would go for lamentably, if someone was trying to evoke sympathy for their plight:

"I am so hungry and I've go no money left. Please buy me some food mister", the boy said to me, lamentably, his big wide eyes pleading with me.

Wretchedly, for me, could be a bit extreme and is for someone really at their lowest, and they may not be trying to evoke sympathy, but rather demonstrating their despair:

"My house burnt to the ground and I've lost everything. What am I to do?" he cried wretchedly.

Or pleadingly... that would seem to be a better option - generally you plead for sympathy.

The two cases are not the same, because persuasion is a verb, while sympathy is a noun. Consequently, while we can say that we were persuaded by a statement, we can't say that we were "sympathised" by it. Instead we would say that the statement aroused our sympathy. We can, however, use verbs that imply supplication, or misery on the part of the speaker; and the natural inference will be that the speaker was trying to arouse our sympathy: "'Please give me some bread,' he implored, 'I haven't eaten since yesterday,' he wept." Alternatively, we can use adverbs that tell us something about the way the speaker conveyed the message: "'I'm awfully hungry,' he said pitifully, 'I can't go on,' he said faintly." All of these can imply that the speaker was trying to arouse our sympathy,

My first thought was ingratiatingly, but Anton's answer is more accurate.

pathetically

In a way that arouses pity, especially by displaying vulnerability or sadness.

Oxford Lexico

having a capacity to move one to either compassionate or contemptuous pity

Merriam-Webster

causing or evoking pity, sympathetic sadness, sorrow, etc.

Dictionary.com

Admittedly I've seen this word more frequently describe actions than speech, but it seems to fit in this context.