Is it permissible for me to give my mother some of my father's money, which is in my possession, without his knowledge, for her to reclaim her right from him and to spend on the household? And would my mother be sinful for doing that?

You are not permitted to give your mother money from your father’s wealth, of which he has entrusted you, because you are a trustee and must safeguard the trust. As for household expenses, if your father is stingy and does not provide you with your basic needs, then it is permissible for you to take from his money, without his knowledge, what is customary. And if your father refuses to repay your mother's debt, and she finds no way to claim her right, then it is permissible for her to take from his money the equivalent of her debt, without his knowledge. However, you are not to enable her to take from the money of which he has entrusted you.