I have had time to look some more, so will add some extra information.
As you have not been back to the thread you started, it seems you may have lost interest, so there seems little point in continuing further.
This 1881 census is the same William WHITE who I have noted in my previous posts (links provided), as the address and age fit. Here his place of birth recorded is just a few miles from the 2 other places he records in the census Lyndhurst and Burley
I believe this is a continuation of the same family and the family you are looking for.
1881 Ringwood, New Street
William WHITE head 49 - Costermonger - born Brockenhurst
Sarah WHITE wife 64 - born Burley
Sarah daughter 17 - glove maker - born Ringwood
The wife Sarah died before the next census, where William is with another wife called Ellen. I dont believe he was actually married to this Ellen. Her husband Lewis LUCKHAM is very much alive.
This is the death for wife Sarah in 1883:
WHITE, SARAH Age 67
GRO Reference: 1883 J Quarter in RINGWOOD Volume 02B Page 430
The daughter Sarah died in 1887.
WHITE, SARAH Age 22
GRO Reference: 1887 J Quarter in RINGWOOD Volume 02B Page 400
Sarah could have given birth to an illegitimate daughter Mary Kate in 1885 the birth rego I quoted earlier would confirm or not.
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Note this marriage
on 3 Aug 1873 in Burley for a William WHITE to a Sarah BARNES, a widow. Her father is Henry SIMMONS.
Sarah SIMMONS is unmarried in 1851 census for Ringwood, living with her sister Rachael.
In 1861, I think she is Sarah Barnes working as a servant in Southhampton.
In the 1871 census in Crow / Ringwood She is Sarah BARNES born 1817 Burley, a widow, unemployed, and boarding with the Browning family.