Daisypetal, Thanks for the clarification between the NY and NJ Markeys.
By noticing in the 1910 census that Patrick and Mary's oldest son Michael was born in Ireland, I was able to find Michael's 1884 civil birth registration in Kells, Meath. From there, I found Patrick and Mary's marriage registration in 1878, also in Kells, Meath.
I now have proximity -- the Markey/Murphy family was in the area where Matthew Lynch came from. But I haven't yet identified the actual connection. I'll leave that to the side for now.
I am waiting for the death certificate from Nottinghamshire in the hopes that it will confirm or reject the premise that Matthew Lynch is the one I am looking for.