What is the ruling on using money given by a father to his son for a specific purpose, to fulfill other needs without the father's knowledge, and is eating from this money considered unlawful?

If you are given money for a specific purpose, it is not permissible to spend it on anything else without the donor's permission, because it is a gift restricted to a particular objective, unless it is known that the donor intended this specific objective but would also be content with other uses. So, if he gave you money to buy, for example, a turban, with the intention of covering your head, it is designated for that. However, if he said it casually, then you may dispose of it as you wish. If the donor intended to free a slave, not as charity, and there remains a surplus of money after freeing the slave, the surplus must be returned to him. If the mukātab (slave who has a contract of manumission for a specified sum) is unable to pay the entire sum, what the master received must be returned to the donor. If the donor intended charity, then no money is to be returned. And whoever is given money for a specific purpose, such as for knowledge, righteousness, or poverty, and that purpose is not present in him, he must not accept it, or must return it if he has accepted it.

Therefore, if your father intended a specific purpose for his money, do not spend it on anything else. If it was merely guidance and he would be content with it being spent on something other than what he specified, then there is no harm. You must be cautious in spending and avoid anything your father dislikes or about which you doubt his approval. In case of doubt, stick to the specified purpose or seek his permission.