It sounds as though you don't have any certificates but are depending just on family stories and what you might have found online?
Birth certificates for the children should list mother's maiden name and (if married) father's name and occupation. Since son John was born in Ireland that would give a starting point for any Irish research.
Possible birth: John McCoy born 2 Aug.1872 Donaghmore district, Co. Down with parents Thomas McCoy and Ellen Campbell. Certificates for other children would confirm Ellen's maiden name.
https://familysearch.org/ark:/61903/1:1:FPBF-NLTAt least one online tree says that Ellen's husband, Thomas McCoy died 1922 Skipton, Yorkshire (born 14 Aug.1848 Ballymacarrraty Beg, Co. Down and married 14 Jan.1872 Glen R.C. Church, Donaghmore Parish)
This seems to be the 1922 death-
https://familysearch.org/ark:/61903/1:1:QVHW-R7S7Possibly him in 1911 in some sort of institution (we aren't allowed to do lookups for 1911 English census)-
https://familysearch.org/ark:/61903/1:1:XWNN-W9VThere are also quite a few possible prison records for Thomas McCoy in the Wakefield area which may explain why he didn't always appear with Ellen and the children.
Added- Donoughmore R.C. parish registers online-
http://registers.nli.ie/parishes/0444