I'll type this out whilst I have the information in front of me - just in case there is some substance to it
1851
1 Dingle Lane, All Saints Poplar, Middlesex
Jonathan DRAKE head mar 30 lab b.March Cambridgeshire
Sarah wife 33 b.Colnbrook Bucks
Thomas son 17 lab b.Folkestone KentDeborah dau 8 b.Oxfordshire
Raphael son 5 b.Poplar Middlesex
Elizabeth dau 1 b.Poplar Middlesex
Emma dau 1 b.Poplar
RG9/310 folio 7 page 8
..........................
1851
Cassington, Oxfordshire
Jonathan DRAKE lodger mar 21 railway lab b.Wymondham Norfolk
Sarah wife 23 b.Coldbrook Bucks
Thomas son 7 b.Folkestone Kentlodging in the household of a WIGGINS family
HO107/1730 folio 503 page 6
A couple of observations:
Jonathan would appear to have been rather young to have had a child born c1843/1844 ie. when he would only have been 13/14 himself at that time - not impossible of course but rather unusual shall we say which would lead me to think that perhaps Sarah had been "married" before - perhaps to the father of son Thomas bc1843/1844 Folkestone
Thomas DRAKE bc1843/1844 Folkestone disappears after the 1861 census - presuming he didn't die could he have reverted to his "birth" name of WHITE?
Jonathan & Sarah DRAKE are in Poplar by 1861 - near to where your Thomas is in 1871 and, furthermore, they have a daughter Deborah as did your Thomas. Ok so Deborah was not a particularly rare name but it wasn't that common either.
There is a marriage of a Jonathan DRAKE with one of the four possible spouses being a Sarah WHITE in the North Witchford registration district (Cambridgeshire) December quarter 1849 ref. 14 258
http://freebmd.rootsweb.com Any thoughts?
Casalguidi