If a mother left a child with a none relative and left state does the father have the right to take the child

Do you mean with a known relative or a non-relative?

The parents are equal rights to custody unless the Court rules (or has ruled) otherwise. Yes, the father can "take" the child, absent a Court Order to the contrary.

For that matter the father can file for custody and the mother (if she decides to pick up the child upon her return) could have a Court battle on her hands.

If BOTH parents are unfit, then someone should step in to protect the child.