I can see the mix of the Foley and Hacket(t) surnames in the births and baptisms for the children, however the closest civil birth I can find for the 2 year old Patrick in that 1901 census shows the mother's name as Kate
Fogarty, John was the informant:
https://civilrecords.irishgenealogy.ie/churchrecords/images/birth_returns/births_1898/02061/1787976.pdf Patrick Hacket was registered on 12 December 1898 (the same day he was born!), but 2 days later he was baptised as Patrick Foley for parents John Foley and Cath. Hackett. All very odd.
I don't see any Foleys in Clashdrumsmith in Griffith's Valuation (local print date 1851), but there was a James Hackett there in a house with a garden.
Clashdrumsmith isn't that far away in several directions from the county border with Limerick, I know the censuses indicate that John and Kate were born in Co Tipperary, but could the marriage have taken place in Co. Limerick? For example, this one in April 1872 between a John and Kate Hackett which took place under 10 miles to the south of Clashdrumsmith:
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1872/11311/8150483.pdf
