Is it permissible to give money of doubtful lawfulness to a girl who is not a mahram, in order to help her since she is poor, and you have advised her to repent and change her situation for the better?

Doubt concerning the permissibility of money does not render it unlawful nor necessitate its disposal. If you wish to be scrupulous and dispose of it out of caution, or if you wish to donate it as charity, there is no harm in giving it to this poor woman, and it does not matter that she is not a mahram to you. You must avoid mixing with her and being in seclusion with her. Inviting her [to Islam] and advising her is permissible within its legitimate Islamic bounds, and it is better to connect her with committed, righteous women so that they may advise each other.