Is taking money from a woman who lives with a non-Muslim man, in an unlawful relationship, forbidden if this money is essential for completing one’s studies, and what is the ruling on maintaining the relationship with her?

You must advise your aunt and warn her of Allah's punishment due to her illicit relationship. It is permissible to shun her if that deters her. As for accepting her gift or tuition fees, it is permissible if her money is lawful, or if it is mixed with lawful money as long as the unlawful does not predominate. However, if her money is solely from an unlawful source, then it is not permissible to take it.