It's usually harder in non children stories where a main character's apology give them direct forgiveness, usually the villain will reject the apology and the other characters may still be bitter and wait for some other amends.

Was reading an old short story breed of battle andwhile it's lightly played for laughs I love how the main character apologize at the right moment for socking the wrong guy in the jaw.
"I reckon I done made a mess of things. I just ain't got no sense when I lose my temper, and I honestly thought you'd stole Mike. I ain't much on fancy words and apologizin' won't do no good. But I always try to do what seems right in my blunderin' blame-fool way, and if you wanta, you can knock my head off and I won't raise a hand ag'in' you." And I stuck out my jaw for him to sock.

He took his hand outa his coat and in it was a cocked six-shooter.

"Costigan," he said, "no man ever struck me before and got away with it. I came to Larnigan's Arena tonight to kill you. I was waiting for you outside and when I saw you run out of the place and jump into a taxi, I followed you to do the job wherever I caught up with you. But I like you. You're a square-shooter. And a man who thinks as much of his dog as you do is my idea of the right sort. I'm putting this gun back where it belongs--and I'm willing to shake hands and call it quits, if you are."