Dear both
Many thanks for these very helpful leads. Like you, I had found him in 1881 as follows:
1881 Census of Stafford
RG11/Piece / Folio   2820 / 102 
5 Churm Terrace 
Walsall Foreign
Henry PONTING   Head   M   Male   36   ...*   Iron Worker     
Eliza PONTING   Wife   M   Female   37   Pelsall, Stafford, England        
Jabez PONTING   Son   U   Male   17   Blakenall, Stafford, England   Iron Worker     
Edward PONTING   Son   U   Male   15   Blakenall, Stafford, England   Iron Worker     
Eliza Ann PONTING   Daur   U   Female   10   Blakenall, Stafford, England   Scholar     
Mary PONTING   Daur   U   Female   8   Blakenall, Stafford, England   Scholar     
Elizabeth PONTING   Daur   U   Female   6   Blakenall, Stafford, England   Scholar     
Henry PONTING   Son   U   Male   3   Blakenall, Stafford, England        
Charles PONTING   Son      Male   9 m   Blakenall, Stafford, England        
*The place of birth is either missing or illegible
From the above, Henry PONTING was born about 1845. I couldn't find any of the children being born in Staffordshire under the name PONTING, but there are several named POINTON, POYNTON or PONTON, which could be a variant spelling.
There is a possible marriage between Henry POINTING and Eliza ROWLEY at Walsall, Staffordshire on 18 DEC 1865.  The question is where was he from?  It has been suggested that he was from Bristol, and indeed there is a Henry PONTING aged 6 (i.e, born in 1845) in the Clifton Workhouse in 1851, but he is the only PONTING in the Workshouse, suggesting that he was abandoned or else his mother had died sometime between 1845 and 1851.
However, your finds suggest Cirencester (or even Wolverhampton!) as possible places of birth.
Curiouser and curiouser...
Happy New Year, by the way, and many thanks for responding.
Best wishes
Phil