Is it permissible to seek forgiveness from one's male cousin (son of paternal aunt) without the knowledge of the paternal aunt, when unable to guarantee something, out of embarrassment from confronting the paternal aunt?

If you do not have the money to compensate for what you have damaged, then the debt remains your responsibility until your affairs become easier or you seek forgiveness from your paternal aunt, because she is the owner of the right, and your responsibility is not absolved except by that. Your seeking forgiveness from her son is not valid, but there is no harm in having him intercede for you with his mother, so that she may forgive you.