Hiya clogdancer,
The Mayo Valuation took place in 1956-7. I would not worry too much about Anthony not appearing in the Valuation. You think he was born about 1836 so his parents may have been born about 1810. They were probably still living when the Valuation took place so the farm would have been in either the name of Anthony's father if he was still living or his mother. There were not too many Maughan's in Mayo so it might be a good idea to see if you can find some information about them and come forward from that time to see if Anthony fits into the picture. You have a couple of possible counties but you need a townland as well.
These are the Mayo Maughans ....
Catherine Maughan, Castletown, Lackan.
James Maughan, Cloonalaghan, Lackan.
Patrick Maughan, Barnhill, Lower Lackan
Patrick Maughan, Barnhill, Upper Lackan
Mary Maughan, Ball- Alley- Lane, Kilmoremoy
Thomas Maughan, Doonadoba, Kilcummin
Micheal Maughan, Beldergmore, Doonfeeny
John Maughan, Rathoonagh, Kilbride
Peter Maughan, Doonadoba, Kilcummin
...... ? Maughan, Sraheen, Ballynahaglish
Best of Luck with your search
Chris