Hi RL
Well, counties Mayo and Galway border each other, so it's possible that the family came from a townland quite near to the border.
If you look at the 1901 Irish Census there are twice as many Naughtons in Galway as to Mayo, so add this to the census actually stating Galway, I suppose I would start there with your searches.
Of course, the downside now, is that you are talking about people born well before our civil registration began, so you need to be looking for church records.
Unfortunately, to do this you need a specific townland in a county to see if church records still exist AND you would need to know what Patrick's wife's name was.
So, have you any leads as to what Patrick's wife's name was ?
Tara