According to MacLysaght's Surnames of Ireland, Horkin is a Mayo variant of the name Harkin. MacLysaght says that the Harkin surname originated in Inishowen (northeast Donegal), with the original Irish form of the name being Ó hEarcáin, from an archaic word for the color red, earc (a word which also has other meanings), with the name thus meaning something like "descendant of the red one". One might think that it refers to red hair, but the word rua is usually used in that case, so maybe it refers to an ancestor whose sking was "ruddy".
One of my grandmothers (a McNulty) grew up in Killasser parish. The McNulty's in Mayo migrated there from Donegal in the 1600's, and perhaps your Horkin's did as well. I've searched most of the parish registers in the area in researching my family history. I didn't note down any Horkin's in Killasser, but then I wasn't specifically looking for them, so they may be there. If there were some still there when the parish register started, and they were not too numerous, you might be able to determine the townland where they lived. If you want to try looking in the Killasser parish records, here's a link to them: https://registers.nli.ie/parishes/0134
If you do locate a likely townland in Killasser, here's a link where you could learn more info about the townland, with links to some other records such as census records and Griffith's Valuation: https://www.townlands.ie/mayo/killasser/
My grandmother's husband (a Gallagher) grew up a few miles away in another Mayo parish. In between the two is the parish of Swinford/Swineford. Records for some of my ancestors are located in the Swinford parish register, either because they lived there or because it had a church building prior to some of the otther parishes, and so people from nearby parishes were sometimes baptized or married there. The records for Swinford go back much further than some of the neighboring parishes. You didn't mention John Horkin's age, but you might try searching there as well. In doing my research, I did note down that a Thomas Harkin was a witness at a Gallagher marriage in Swinford in 1826.
I just did a quick search for Horkin's in the Family Search database, and there were several Horkan families shown as living in the area between the town of Swinford and the town of Foxford in the 1860's (an area overlapping the parishes of Swinford and Killasser, and also the parish of Toomore). Don't be put off by the spelling of the name, by the way. Different anglicized forms were written down by the authorities and parish priests at various times, and the spelling can vary from generation to generation (sometimes even in the same generation) within the same family.
If you want to try looking in the Swinford parish records, here's a link to them: https://registers.nli.ie/parishes/0133
If you've had your DNA tested, we could compare results. Given the Killasser connection, we may be at least distant relatives. My results are posted at GEDMATCH, as kit number T780556. It still might be hard to find a paper connection, of course, although I and some relatives are trying to track down the estate records for the area (the landlord's records), which somtimes go further back and can contain useful genealogical info.