You may have to pay back welfare, regardless of whether she defrauded them or not. It is very rare that they will prosecute a mother for fraud when they can go after the father.
There is a case in Modesto where a father has always had custody, for 16 years. The mother kept filing for welfare/AFDC and getting it. But, rather than prosecute her, they went after him. He currently owes over $25,000.
And if he smart he will make sure he doesn’t pay a dime and forces the government to declare fraud on the mother.
Will this happen – no because:
1) the false belief fathers are paychecks and
2) no one wants to rock the boat or the gravy train.