This may be clutching at straws but there's an interesting entry in the 1881 census.
District West Derby Workhouse, Walton on the Hill, Lancashire
McCarthy Maria Wid 31 Occupation None, Born Ireland
McCarthy John Unm 5 Occupation Nonde, born Lancashire Liverpool.
RG11/3694; Folio: 56; Page: 17
As these two follow one line after the other in the entry, I would suspect that this is mother and son.
Now there are records at the Liverpool Record Office for this
RefNo 353 WES
Title Statutory Authorities: Boards of Guardians. West Derby Union
Date 1848 - 1945
Description Board Minute Books, minutes of various committees, account books, newscuttings, weekly returns of persons relieved, records of Walton Workhouse, Chaplain's Report Books
and you might want to read a little
http://tinyurl.com/rsw7hThe entry for these people may state John's father's name and so could possibly confirm whether it is him or not.