Is the mother obligated to pay the remaining amount of a vow (100 Egyptian pounds) after the father refused to pay the full amount, or is the father's statement correct and she is not required to do anything?

If her vow is contingent on the recovery of her son, it is dislikable [makrooh], but she is obligated to fulfill it if her son recovers, based on the saying of the Prophet, peace and blessings be upon him: "It does not bring forth any good, but rather it is used to extract [something] from a miserly person." However, if she made an absolute vow without any condition, she is also obligated to fulfill it. And when she is obligated to fulfill it, her father does not have the right to prevent her from doing so, unless she vowed more than one-third of her wealth, in which case he can prevent her according to Imam Malik, contrary to the majority of scholars. And since your father has paid 400 Egyptian Pounds on her behalf, she is obligated to pay the fifth hundred, and it should be spent on what she specified in her vow. If she did not specify, it should be given to the needy.