Is secretly sending $50 to one's mother to cover her expenses, beyond what the father provides for her, considered permissible or forbidden?

You are not permitted to take anything from your father's money that he entrusted to you, except with his permission. Your mother's situation has three possibilities: If what your father gives her is sufficient for her, then she is not allowed to take from his money without his permission, and you must advise your father to be just. If what he gives her is not sufficient for her, then she is permitted to take from his money in moderation without excess. If she is unable to take her right herself, then you are not permitted to take from your father's money, and you must advise him and fulfill your agreement with him. Her maintenance then becomes a debt upon him, and she can demand her rights through the judiciary or mediators, and you all must treat her kindly.