Legal Question in Family Law in New Mexico

Re: child visitation after father abandoned child

First you want to ask him for the child support arrearages. Tell him how much he owes you and demand payment. If he doesn't pay then you go to court and ask for relief. You can get the court to attach his wages and put liens on his property. They can find him in contempt and jail him. If you have been on state assistance of any kind CSED can help you with this.

Courts tend to treat support and custody/visitation separately, but his failure to pay suppoprt will be taken into account when considering visitation.

Your question about abandonment: abandonment is a ground for termination of parental rights, which you may be able to get done. Although if he's back and persuades the Judge he wants to participate in the child's life the court may allow it because abandonment is based on what the Judge sees in the future. It's a judgement deal for the court but you've got a shot at it. You ought to get a lawyer to navigate this thicket.