As an illegitimate child called Hosthead and brought up in the Hosthead family, if none of that family were prepared to look after the child she would go straight into the workhouse. She wouldn't be suddenly dumped on the Robinson family (as their responsibility - which she legally wasn't) and acquire a Robinson surname - which legally she had no right to (though you can and could be known by any surname you chose), and then subsequently be placed by that family into the workhouse system. It is extremely rare that an illegitimate child's paternal family take up the responsibility for the child, unless a marriage at some point occurred between the parents.
Births don't get registered and people do get missed off or wrongly enumerated on censuses. It is perfectly possible that Dorothy Robinson has falled through both those cracks - no birth registration and the whole 'family' including her father missed off the 1851 census.
Surmising anything about the surname Robinson is very tricky because it is such a common surname. The 12th most common surname in the country, so you easily get coincidences like this even in Gainsborough and even with an extremely rare surname like Hosthead which I'm not sure even exists anymore.
ROBINSON, Thomas
Susannah HOSTHEAD
Marriage Date:31 Oct 1849 All Saints Gainsborough, Lincolnshire
Husband's Father: Thomas ROBINSON
Wife's Father: William HOSTHEAD
These Robinsons moved to Durham and from the later censuses it is apparent this Thomas was born in Derbyshire.
FreeBMD is of course incomplete for the period we are interested in so any of these Dorothys mentioned could have further entries in the civil registration - not just for the Gainsborough registration district - which doesn't just cover Gainsborough.
Gainsborough Registration District
Parishes in Lincolnshire (1837-1930):
Blyborough, Blyton cum Wharton, Brampton, Coates, Corringham, East Ferry, East Stockwith, Fenton, Fillingham, Gainsborough, Gate Burton, Glentworth, Grayingham, Greenhill and Redhill, Hardwick, Harpswell, Haxey, Heapham, Hemswell, Kettlethorpe, Kexby, Knaith, Laughton, Lea, Marton, Morton, Newton on Trent, Northorpe, Owston, Owston Ferry, Pilham Scotter, Scotton, Southorpe, Springhtorpe, Stow, Sturton by Stow, Thonock, Torksey, Upton, Walkerith, West Butterwick, Wildsworth, Willingham, Willoughton.
Parishes in Nottinghamshire (1837-1930):
Beckingham, Bole, Misterton, Saundby, Walkeringham, West Burton, West Stockwith.
All I know from the civil registration is these other possible Dorothy's had their births registered in Gainsborough registration district. I don't know that they were actually born in Gainsborough. e.g. 1851 census
Misterton Nottinghamshire
Dorothy Berry 1 Misterton, Nottinghamshire, Daughter of Thomas
and the Dorothy Green death registered in Gainsborough registration district in 1853 is possibly this woman.
West Stockwith Nottinghamshire
Dorothy Green 70 Harworth, Nottinghamshire, Head Widow
Gainsborough parish registers for baptisms have coverage on the IGI only until 1835, so whether Dorothy 'Robinson's' baptism did or didn't occur in the Gainsborough parish register is still unknown.
Regards
Valda