Is it permissible for a mother to give her poor maternal aunt some of the family's provisions (flour, oil, and sugar) without the knowledge of her miserly husband?

It is not permissible for a wife to dispose of her husband's money by giving charity without his explicit or customary permission. Therefore, it is not permissible for your mother to give charity to your aunt from your father's money without his permission. As for your mother's own money, she may dispose of it as she wishes without needing your father's permission.