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I'm not from Mayo, but did a little searching.  Wondering if this might be your Halligan family in 1901 & 1911?

http://www.census.nationalarchives.ie/pages/1901/Mayo/Burriscarra/Rinan…

http://www.census.nationalarchives.ie/pages/1911/Mayo/Burriscarra/Rinan…

There are two James Halligans in the Griffiths for Burriscarra, one in Rinaneel:

http://www.failteromhat.com/griffiths/mayo/burriscarra.htm

Looks like the children were born in Hollymount.  I didn't find any Halligans there, but I did find a John Farragher in the 1901 & 1911 censuses.  Maybe you have some info that might prove this to be accurate or not?

http://www.census.nationalarchives.ie/pages/1901/Mayo/Hollymount/Skehee…

http://www.census.nationalarchives.ie/pages/1911/Mayo/Hollymount/Skehee…

Found a John Farraher in the Griffiths for Skeheen and Lisseveleen:

http://www.failteromhat.com/griffiths/mayo/robeen.htm

If you can confirm these townlands to be correct, hopefully someone local can get you back further or connect you to family in Ireland.

Cathy

Cathy, thank you very much for your assistance. The John you found may well be the son of 'my' John Farragher.

From my calculations The James and John I am after would have been born around 1815 to 1820 (assuming marriages between 20 to 25 years of age, and children shortly thereafter). Hollymount, Burriscarra and Rinaneel are all names I recognise from my research.

The Marriage Record of Stephen and Bridget 14/1/1864 shows Bridget's address as Fraispark.

Wondering if Fraispark could be Friars Park?  That being said, I can't find in Mayo.  I see one in Meath and one in Roscommon.  I couldn't find anything for Frais Park.

Cathy