Is it permissible to count money spent on family as assistance, as a form of Zakat, given the presence of the father who does not provide for them?

Zakat (definition) is not permissible for family members whose maintenance is obligatory upon you (parents, wife, and children). As for other relatives, such as brothers and sisters, it is permissible to give them from Zakat if they are poor and needy and do not earn an income, provided that they take possession of it and dispose of it freely, and that this does not prevent what you used to give them previously.