If a woman gets child support by fraudulently claiming the child lives with her, can she be ordered to repay the father?

A good father had his child living with him, and was the sole support, as the mother neglected the child and would not even provide basic care. The mother obtained welfare benefits and child support by claiming the child lived with her. The father is someone who did not stand up for himself, and thought he had no choice, so paid out in excess of $20,000 in addition to caring for and supporting the child that lived with him. (The child is now grown). Now, a few years later, the father has seen how wrong this is, and wants to know if there is any way to make her repay him for what she obtained by fraud?