Who has the most right to receive the reward that children get from their schools: the children themselves, or their mother who demands a share of it, or the father who relies on the saying, "You and your wealth belong to your father"?

The money given to children is their property, and the father has guardianship over the wealth of his young children, so he receives it and disposes of it in their best interest. It is permissible for parents to eat from their children's money, provided there is a need and no extravagance; due to the Prophet's (peace be upon him) saying: "You and your wealth belong to your father," and this is for permissibility, not ownership, and it includes the mother as well.