The most promising record for Edward, son of Ann Monaghan is the 1881 census record showing James Goolden, Ann Goolden and Edward Monogue.
RG11 Piece 4036 Folio 51 Page 15
This gives Ann's year of birth as 1845 and place of birth as Ireland.
Checking Irish birth/baptismal records for 1840-1850, I have only found two possibles.
Anna Monaghan baptised 1844, daughter of Thoma Monaghan and Brigida Brien, at Saggart on the edge of Dublin.
Ann Monaghan baptised 24/7/1842, daughter of Thomas Monaghan and Bridget Monaghan, at Kildress, Tyrone.
I think it's time to step back, gather together all the records and review what we have. Thanks.