I can see the family in the 1841, 51 and 61 censuses of England, but not 1871.
Any sign of them in 71?
In 1841 ages were rounded down to the nearest 5 years. With that in mind, I see
John Hayes - 30 born Ireland, i.e 30-34 years old, tailor
Elizabeth " - 20 born Ireland, i.e. 20-24 yo.
Mary " - 2, born Lancashire.
George " - 20, born Ireland, i.e. 20-24 yo, tailor.
This George must be a brother of John.
John's age makes it difficult for him to be the son of the George Hayes who died in Coolrainey in 1875. Also this George in Liverpool makes it difficult for John to be the brother of the George in Ireland?