The 1871 Census has the mother called Bridget and these marriages suggest the father was called Andrew:-
Name: Margaritam Heraghty
Marriage Date: 15 May 1861
Parish: Our Lady of Reconciliation de La Salette, Lancashire, England
Spouse's Name: Patricium Fleming
Father's Name: Andreas Heraghty
Spouse Father's Name: Joannis Fleming
Name: Thomam Heraty
Marriage Date: 27 Feb 1867
Parish: St Mary, Lancashire, England
Spouse's Name: Elizabeth Manning
Father's Name: Andrai Heraty
Spouse Father's Name: Joannis Manning
So possible match for Thomas' birth/baptism:-
Name: Thomas Heraghty
Year: 1843
Father: Andrew
Mother: Bridget
Parish: Aghagower/Aughagower RC
County: Mayo
http://www.rootsireland.ie/
Griffith's Valuation of Ireland, 1847-64 (this record was printed in 1855)
Occupier: Andrew Heraghty
Lessor: Sir Wm Roger Palmer Bart
County: Mayo
Barony: Burrishoole
Parish: Islandeady (next parish to Aghagower/Aughagower)
Townland: Carrowbeg
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