Is it permissible for the questioner or for his mother to borrow from the usurious money that the questioner got rid of by giving it to someone who lends it to others without charge, when there is a need for that?

You have done well in disposing of unlawful wealth and spending it on public Muslim interests, such as the poor and the needy. It is permissible for you or your mother to take from it according to your need if you meet the description of poverty, and it will be lawful and good for you.