I have been to the website you mentioned, thank you. It is impressive. I have come up with either Tawnagh Beg or Tawnagh More as the most likely contenders. They are both in the Barony of Gallen, and the Civil Parish of Templemore.
My great grandfather Patrick Neary was born in 1842, my grandmother in 1882. She married James Leo Harkins from Scotland, they married on 29th July 1907, and they had 8 living children together, 7 daughters and my father, who was named after my grans brother Martin. My dad was born in 1918, his name was Martin Francis Harkins, but went by Frank.