What does ヘニャッ mean in the following sentence?
As with any mimetic word in Japanese, the meaning is strongly connected to how one would imagine an action would sound like. Considering the context of "wanting to be spoiled or pampered", it can be related to letting go of all muscular control and folding up or lying flat like a sheet of dough (as opposed to something that has bones and joints).
If we were to extend the sound to ヘニャァァ or something similar, then it would conjure an image of something slowly spreading outwards, like a gel -- a more emphatic form of the action discussed in the previous paragraph.
I agree with Chocolate that it is 擬態語. Thus if I understand correctly it is a sibling of onomatopoeia, but not quite the same thing, because ヘニャッ doesn't come from sound.
The mental picture this word should evoke is something soft that buckles or collapses. So in the context of what you provided, I think it's a (tired) human body that buckles (onto the lap of the girl/boyfriend or something like that I'd imagine, since you say 甘えたい)