I am slightly confused by the information you have given. You start by saying that the family may have come from county Omagh. Omagh is a town in Co. Tyrone. It’s the county town. On the other hand we have a Co. Armagh, which is pronounced much like Omagh. Did you perhaps mean Armagh?
Then you say that you think William McConnell was born in Cork (which is at the other end of Ireland from Omagh and Armagh), and his wife Mary Clendenning was born in Couny Maugh which, to me, might be Connemara, which is a region in Co. Galway in the far west of Ireland. (Which makes me wonder how husband and wife would have met).
Can you clarify this? Also, what denomination were the family?