If the father of my newborn knowingly left me homeless during pregnancy and now wants overnights with newborn, do I have a case for neglect?

In Georgia, for unmarried parents, unless and until the father legitimates the child and obtains court ordered visitation or custody, the mother has full custody of the child. You can give him no or as little visitation as you want. Full custody is with the mother regardless of whether you seek and are awarded child support or not. Therefore, you should consider seeking child support. Nevertheless, if the father wants to establish a relationship with his child, you may want to consider encouraging one. With only a few exceptions, it is in the best interest of the child to have some contact with the father, even if it is just supervised visitation.