Dear {Recipient},
I would like to apologize to you for {incident}. Everyone makes mistakes, and I know that doing that was wrong. I know that you are angry, but I hope that this apology will help you see that I am very sorry.
To make it up to you, I would like to {restitution offer}. If that sounds like a good idea, we can do that on {date}.
Again, I am very sorry.
Yours,
{Sender}