Do you think Jaime losing his hand is some sort of poetic justice for when he tried to find Arya so he could cut her hand off? (Though IMO Jaime should have suffered much more, he tried to kill one child and hurt another, he deserves the worst treatment there is)
Yes and no, but mostly no.
They took my sword hand. Was that all I was, a sword hand? Gods be good, is it true? (Jaime IV, ASOS)
They had taken his hand, they had taken his sword hand, and without it he was nothing. The other was no good to him. Since the time he could walk, his left arm had been his shield arm, no more. It was his right hand that made him a knight; his right arm that made him a man. (Jaime IV, ASOS)
“I’ve lost the hand I killed the king with. The hand that flung the Stark boy from that tower. The hand I’d slide between my sister’s thighs to make her wet.” (Jaime V, ASOS)
Yes in that wanting to maim Arya would definitely fall into the category of “the hand that flung the Stark boy from the tower” in terms of “villainous deeds performed by Jaime Lannister which were symbolized by that hand and which now are calling him to reexamine his place in life”; but no in that he didn’t actually do it–unlike throwing Bran out a window, doinking Cersei, or killing Aerys.
If he had actually cut off Arya’s hand or killed her the way that Cersei wanted, that’d be a whole different thing. But he doesn’t think of the Arya situation until a book after he loses his hand, which makes it feel–from an authorial angle–like an afterthought compared to those three mentioned above which happen right away.
“Do you see that window, ser?” Jaime used a sword to point. “That was Raymun Darry’s bedchamber. Where King Robert slept, on our return from Winterfell. Ned Stark’s daughter had run off after her wolf savaged Joff, you’ll recall. My sister wanted the girl to lose a hand. The old penalty, for striking one of the blood royal. Robert told her she was cruel and mad. They fought for half the night… well, Cersei fought, and Robert drank. Past midnight, the queen summoned me inside. The king was passed out snoring on the Myrish carpet. I asked my sister if she wanted me to carry him to bed. She told me I should carry her to bed, and shrugged out of her robe. I took her on Raymun Darry’s bed after stepping over Robert. If His Grace had woken I would have killed him there and then. He would not have been the first king to die upon my sword… but you know that story, don’t you?” He slashed at a tree branch, shearing it in half. “As I was fucking her, Cersei cried, ‘I want.’ I thought that she meant me, but it was the Stark girl that she wanted, maimed or dead.” The things I do for love. “It was only by chance that Stark’s own men found the girl before me. If I had come on her first…” (Jaime IV, AFFC)
Even that memory is more about Cersei than about Arya. Arya is routinely a tangential afterthought in Jaime’s mind rather than a focal point, which I think makes sense given the main sources of “Stark Tension” in his arc–Bran’s attempted murder and the quest to find Sansa. He, like most people, assumes that she’s dead–but more than that he only thinks about her either in connection to Catelyn’s bargain or her likely being dead. Even when he meets Jeyne Poole pretending to be Arya, his reaction is disconnected from any kind of weight Arya may influence over his personhood–unlike Bran, Sansa, Aerys, and Cersei.
Jaime studied her closely. “You know me, then?”
She bit her lip. “You may not recall, my lord, as I was littler then … but I had the honor to meet you at Winterfell when King Robert came to visit my father Lord Eddard.” She lowered her big brown eyes and mumbled, “I’m Arya Stark.”
Jaime had never paid much attention to Arya Stark, but it seemed to him that this girl was older. “I understand you’re to be married.”
“I am to wed Lord Bolton’s son, Ramsay. He used to be a Snow, but His Grace has made him a Bolton. They say he’s very brave. I am so happy.”
Then why do you sound so frightened? “I wish you joy, my lady.” Jaime turned back to Steelshanks. “You have the coin you were promised?”
“Aye, and we’ve shared it out. You have my thanks.” The northman grinned. “A Lannister always pays his debts.”
“Always,” said Jaime, with a last glance at the girl. He wondered if there was much resemblance. Not that it mattered. The real Arya Stark was buried in some unmarked grave in Flea Bottom in all likelihood. With her brothers dead, and both parents, who would dare name this one a fraud? “Good speed,” he told Steelshanks. Nage raised his peace banner, and the northmen formed a column as ragged as their fur cloaks and trotted out the castle gate. The thin girl on the grey mare looked small and forlorn in their midst. (Jaime IX, ASOS)
Jaime says it himself: he’d never paid much attention to Arya Stark. Maybe that’ll change in coming books but I think it’s misreading his arc and the symbolism of his hand and subsequent identity crisis to say that it’s poetic justice for Arya more than, say, killing Aerys.