My grandfather grew up about a mile from Kinaffe/Kinaff (in Irish, Cionn Damh), and it's not actually in Aghamore parish, but across the parish boundary in Kilconduff (Swinford) parish.
You can see more info about it here: https://www.townlands.ie/mayo/gallen/kilconduff/killedan/kinaff/
There are links there to the Griffith's Valuation records for the townland, and there is a John Murtagh shown, but no Patrick. Links to info for the neighboring townlands are also shown, and you can see that the townland just to the west of Kinaffe, Knockbrack, is in the neghboring parish of Bohola. Three parishes come together in that immediate area: Swinford, Bohola, and Killedan (Kiltimagh), and most people would have had relatives from all three parishes (and Aghamore, which is nearby as well).
The Catholic parish records for Swinford go further back than any others in the area (1808 for marriages and 1822 for baptisms), and they are available online here:
https://registers.nli.ie/parishes/0133
I did a quick check, and the baptism of your James Murtagh is there. He was baptized on 8 August 1868, and his parents are listed as Patrick Murtagh and Bridget O'Brien (with their first names all "Latinized", as some priests chose to do), and their townland of residence is listed as Kinaff. I'm attaching a copy of the entry below (it's the first one on the first of the two facing pages). The civil registry had been created only a few years before, and people were not entiirely used to it. There was a fine for registering more than a certain number of days after the birth, so people would give a false birth date if they were late in registering (the authorities would not bother to consult the parish register to check, since it was not offically accepted for a period of time).
I have both Murtagh and O'Brien relatives myself, but know of no specific connection to your family. However, some of the Gallagher and McNicholas tenants listed in Kinaffe in Griiffith's Valuation are very likely to be my relatives, and most of the families had been in the area for centuries (some perhaps for thousands of years), so we are probably at least distantly related. I've done DNA testing, in case you want to compare notes.
In case you're interested in history, there were two Irish surnames (from unrelated families) which were both Anglicized as Murtagh (sometimes Murtaugh) or Murtha. They were the surnames Ó Muircheartaigh ("descendant of the navigator") in Leinster and Mac Muircheartaigh ("son of the navigator") in Roscommon (which borders Mayo nearby, and is probably the main source of the name in Mayo). Actually, there may have been three families, since the name was also found in Scotland, and brought from there to Ulster, where it was usually Anglicized as Murdoch. Either of the two Irish forms of the name could end up as Murtagh or Murtha, depending on how they were heard by whoever was writing out the Anglicized forms (Irish has a number of sounds which English speakers find hard to hear clearly, and there are also regional difference in pronunciation). It was thus not unusual for different Anglicized versions of a surname to be used within the same family, and in fact the Murtagh relatives I have, many of whom ended up in Missouri in the US, use both forms in different branches of the family. What may also have contributed to the discrepancies is the fact that they would all have been Irish-speaking in the 1800's (my grandparents grew up speaking Irish at home in Mayo even in the early 1900's), so they may not themselves have known (or possibly cared all that much) how the name was written in its Anglicized form, at least until later on in the century or when they emigrated.
In case you're interested in folklore or local color, Dublin City University has a huge online collection of folklore from most parishes, collected in the 1930's by local schoolchildren, and the collection covers all sorts of topics about local history, people, place names, etc. There is interesting info for all of the parishes mentined above, and you can view the records here: dúchas.ie
