Is the money that the beggar took from donations and used to pay off part of his debt and household expenses considered unlawful (haram), what is the ruling on feeding his family and children from it, and how can he absolve himself of that?

If you were given money to distribute to the needy and you yourself are in need, it is permissible for you to take from it according to your need, and nothing is required of you. However, if you took from it while you are not among the category intended by the donor, then this is forbidden, and you must repent and return the money to the donor or the specified entity without informing them of what you did.