According to the Civil Code, biological parents have the obligation to support their children, regardless of whether they have a marriage certificate or not. Children born in wedlock and children born out of wedlock enjoy the same rights in law. If the woman refuses, she can sue the court for child support. It is not a problem that there is no household registration. The medical records of the time when the baby was born can be used as evidence. If it is not feasible, parent-child identification can also be done. You can bring a lawsuit to the court in the name of the dissolution of the cohabitation relationship, and judge the custody of the child at the same time.