I'm afraid the 1871 doesn't help as far as a father is concerned.
RG10/2407/37/29
Lye Water, Crewkerne
Sarah Bargery, 35, Head, Widow, Glove Sewer
Eli, 15, Factory Hand, Flax
Henry Charles, 12, Ag Lab
Sarah Ann, 8
William 6
Mary Jane 1
There is a marriage in Dec 1852 Chard, 5c/863 - Eli Bargery, ( possible spouse Sarah Mills or Hannah Brewer) This would fit the eldest son's name. There is also a death for Eli Bargery, 1864, Taunton 5c/272 but if that is actually Sarah's husband, who is the father of the youngest child!
Sarah is obviously unclear about her own age and the ages of her children - there is quite a discrepancy bteween this census and 1861.
I think I'll have a look for Eli Bargery and maybe an unmarried Sarah bargery in 1851. It is entirely possible that she was never married.
Hang on - I'll be back....
Jen