Can a father in Texas,relinquish their rights to a child just so that they don't have to pay child support?

Did you ask the same question this morning using another user name ?

But no they can not do it, a father can not just sign over his rights, and most certainly just to get out of paying child support

See this post

https://www.askmehelpdesk.com/family-law/signing-over-rights-read-first-116098.html