My gut instinct here is that Harriett was indeed born Farmer - this maiden name given for daughter Harriett Hilda born in Brighton 1910.
She states in 1911 (as already stated) that she was born in St. Pancras and in 1939 as born 12/4/1881. The birth shown as 13/4/1881 St. Pancras Workhouse (posted by wivenhoe), illegitimate daughter of Harriett Farmer sure sounds good to me.
I suspect the 1891 family in Deptford is probably Harriett, young Harriett and now supposedly married to a John Carroll. Cannot find a marriage between John Carroll and Harriett, nor find them together after 1891. Suspect they were never married at all which can explain why Harriett would be confused between Farmer and Carroll as being her true surname. Her earlier children could have been fathered by a 'Mr. Mitchell' - the bit that throws me is her marrying Arthur Savage with the surname 'Hanns'??
Can you please, as has been asked already, give us all the details that are on the 1909 marriage certificate?
Annette