I am wondering if Hannah was not actually born in Durham but lived there as a young child, as I can't find any birth registration under any variation starting F or T. 1840+/-2
There is a Bridget Farney age 19 born Ireland working as a servant in 1851 for an Elizabeth Clark at Crossgate Durham. Older sister or relation?
Irish names are often terribly misspelt in records, which may explain all the different spellings.