Is receiving financial aid based on incorrect information about the mother's employment considered unlawful gain, and how can one dispose of it if unable to return it?

Informing them that your mother is not working, when she is, constitutes a lie. If the aid requires that the mother not be working, then it is impermissible to obtain it. You must ascertain whether the mother's employment is a barrier to receiving the aid. If it is, then what you took must be returned. If the grantor is a private entity, the money must be returned to them. If the grantor is a public entity and you know that they will spend it on its legitimate uses, then it must be returned to them. Otherwise, it should be spent on general Muslim welfare.