In the book and the show things don't go too well for the Hound. In both cases, he gets horribly wounded and asks Ayra to kill him. However, she does not. Why is this? What message was she trying to send? All that Sandor is greeted with is silence. I thought it was a tad bit odd since originally the Hound was on her list of people to kill.
The message Arya was trying to send is that the Hound doesn't deserve a clean, merciful death.
In my opinion, this action foreshadows Arya's future path in Braavos:
She is trained in the House of Black & White, the temple of the Faceless Men. The Faceless Men call mercy kills "the Gift". Without her knowing yet, Arya is denying the Hound this "gift".
In Chapter 74 of A Storm of Swords Arya tells him
"You don't deserve the gift of mercy."
There may be more to it, but reasons she gives at the time are
She never forgave him for killing Mycah, or for hitting her with an axe instead of trying to save her mother at the Red Wedding.
I have only watched the show, but to me it is clear that she is conflicted in many ways about killing the hound at this point.
Firstly, she wants him to suffer, he killed the butcher's boy as well as the farmer (who he nicked the silver coins from) and therefore she hates him. However, he has also been her protector to a degree and she likely knows that she owes him her life to some degree while at the same time being his captive. Lastly, I think she is still a young girl and while she has killed a few folks here and there, she has never killed someone who she has a relationship with.
In the end, I think she chose the easiest way out for herself, letting him die in the wilderness by lack of action on her behalf.
"It's pretty firmly established that she did have him on her list." – Monty129 Jul 9 at 0:14
I agree it was clearly established that the Hound was part of Arya's list, which is why NOT killing him was the only way to convey with certainty that she had removed him from it. Whether she feels he doesn't deserve her mercy or whether she simply can't bring herself to kill someone to whom she has grown close is far less evident. I personally believe she posses the strength of character to deliver mercy, therefore I suspect that, while she has developed an affinity for the Hound, she still harbors resentment about his past cruelty and is unwilling to intervene.
Arya developed a conflicted relationship with Sandor. He taught her how to survive on her own. He gave her the common sense and necessary evils she needed to stay alive but she still had a heart. As did the Hound. I think she left him, took his silver, etc to make a point.
When they left the farmer, they were arguing... the Hound was convinced that he would die.
Sandor: Aye, he's a good man. His daughter makes a nice stew. But he's weak. They'll be dead come winter.
Arya: You don't know that.
By leaving him, she KNEW he wouldn't die. But by taking his silver, she sent a poignant message.
That just because he is gravely wounded and "weak"... he won't necessarily die. Which is why the Hound broke his own "code" and took the farmers silver. Arya knows that Sandor has good in him. And that's the reason he "will never be a real killer".
When his silver was about to be taken by Arya, he tried to take it back. He had said in a previous episode that "dead men don't need silver". When Arya saw this, she knew that the Hound still has hope that he will still live. She decided to just leave him there without taking away that hope.
This is just what I think. I've only seen the TV show and I don't think TV show Arya can resist caring for the Hound even just for a little bit.
I think it's both. On the surface (and what she shows to him in her behavior and words), Arya is leaving The Hound to suffer and bleed prefusely, and die on her terms - devoid of mercy, similiar to how he treated his victims. A part of her despises him, his previous malicious actions and what he stands for. He killed her friend, served the Lannisters, failed to save Catelyn and Robb at the Red Wedding, he had her captive and up for sale - and he is an early name on her "To Kill List".
On the other hand, deep inside she has developed feelings of friendship and gratitude towards him as well, knowing he saved Sansa from rape, he despises Joffrey and the Lannisters, he basically saved her from her mother and brother's fate at the Red Wedding, he was her companion and some kind of a guide to her (teaching her mostly violence, naturally for Game of Thrones), and in a way she now owes him a debt which is problematic because he's on her list.
In my opinion, it's not that she couldn't do it morally... It's that she didn't want to. And she was convinced he was dying in agonizing pain anyway. Watch Season 6 knowing all this... 😉
I definitely feel like she didn't kill him because they grew close. He's one of the first people to be on her list, and she has plenty of opportunities to kill him. Everyone else she comes across who happens to be on her list meets their demise. But she won't kill The Hound because he's not on her list anymore. Although, she doesn't feel like she needs to help him either.
By not killing him Arya is conveying that she has forgiven him for his past crimes. Killing him would have crossed him off her list and thus the 'forgiveness' would have been hard to show. Yes she could have said it with words but that would have been 'weak' and the Hound would have felt shame by doing it.