I wonder if my Tarpey puzzle might connect with yours - I'd be interested to know the details of any Tarpeys you have.
My gg-grandmother Sarah Tarpey was born illegitimate in Huddersfield in 1845 (the birth was registered as Tarp). Her mother was Mary Tarpey, born about 1821 in Sligo. Mary does not appear in the 1841 census for Huddersfield (it is searchable online), but they appear together in the 1851 census. Mary married Lewis Landy, a Jewish Hungarian in 1852 (she described herself as a spinster) and had some more children Ann, Mary, Esther and Joseph. She died in 1879.
Sarah married Peter Galvin in 1864. They had children James, Mary, Sarah Ann, Luke, Peter and Bridget.
No father is given on Sarah's birth certificate, but on her marriage cert, father is given as "Edward Tarpey", occupation "Deceased". Now this may be fictitious, or her father may have been called Edward but not Tarpey. I suppose there's just a long shot he may actually have been called Edward Tarpey (e.g. if her father was her mother's cousin).
On Mary's marriage certificate she gave her father as "James Tarpy, deceased", a farmer.
I've hit something of a dead end with them, as I can't find - and don't really expect to find - a birth for Mary. But I suppose it's possible they are somehow related to your branch - Huddersfield is fairly close to Leeds, and Mary had no disernable relatives in Huddersfield - there are no other Tarpys in any Huddersfield census (1841-1901 are all searchable online).
It's certainly something of a mystery this lone Irish girl of about twenty is suddenly parachuted into Huddersfield with no relatives identifiable, sometime between 1841 and 1845 (potato famine might well be to blame)