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

Solution 1:

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

Solution 2:

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.

Solution 3:

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