Hiya,
In the light of ages, etc., it had to be Robert Mansfield senior who was the pub landlord in 1855.
On FreeBMD, I've found a death of a Robert Mansfield in Newcastle upon Tyne in 1858, which I assume you can tie to the certificate you mention. Does that give any clue as to where he was born to try and track him in 1841, find his parents, etc.?
Neither FreeBMD nor IGI throw up the Robert & Mary marriage, although they seem to agree that Harriet was born in Woolwich & christened there 21/2/1847, and FreeBMD has Robert junior in Leeds in quarter to March 1849.
From findmypast (for which I don't have a subscription, just prepaid credits which I must admit I'm just using for my own research):
* I can see Robert junior in Woolwich in 1871, 1881 (with mother at 41 King Street) and 1891 (by then in an institution - can't tell exactly which).
* Can't see Mary after 1881, nor Harriet after 1861.
* 1851 isn't complete, so lack of any of them proves nothing much.
* Whilst there's a couple of Mary Mansfield's born 1811 showing in 1841, they don't seem to be the right one, but there are a few Robert possibilities.
I have to say I can't see any evidence of Robert junior marrying or having children.
FreeBMD has several Harriet Mansfield marriages, but I can't see how to push it any further unless you know her married surname.
Hope this helps a bit in your search.
Phil