That Edward Corkwell (1871 Basford census) was born 1790 and his occupation is agricultural labourer and he is an inmate at the Basford Union Workhouse.
The Edward you are looking for was a hand loom weaver.
If you cannot accept that your ancestors were illegitimate, perhaps it would be better not to pursue genealogy.
Look for bastardy records, see if Esther named the father of Fanny, the question would have been asked if she needed money.
Times were hard, Esther's age is 27 in 1851 which means she had Fanny at age 17/18. She moved in with a man twice her age who may or may not be the father of Fanny. They stayed together for 20 years.
Something happened after Thomas was born 1864 he was registered as Corkwell, yet one year later
Esther marries the lodger Henry Brown,
In 1861 Edward was already 65 perhaps he could no longer provide for the family due to illness or affliction.
Look for a death circa 1861-1865 see if you can find Edward Corkwell in the area.
Her sons John and Thomas may belong to the lodger and not Edward.