Is it permissible to accept financial aid from a non-mahram man to pay off a debt, knowing that Sharia does not approve of any relationship between a man and a woman outside the framework of marriage?

There is no harm in accepting money from a non-mahram (definition) man as a gift, for gifts are generally permissible. The Prophet, peace and blessings be upon him, used to accept gifts and reward for them. It is permissible between a man and a non-mahram woman. Nevertheless, one should fear Allah and cut off the relationship with this man. If there is a need to speak with him, it should be limited to that need, while adhering to Islamic manners and ethics in dealing with a non-mahram man.