Hello Maureen
Most intriguing ! I wouldn't worry about the birth certificate scan being deleted because I think that it's illegible in that tiny format. The most important thing that you confirm in your message is that the baby whose birth registration relates to that certificate was born at the same address where "Mary McGain, 30 yr old married lodger, b Wolverhampton" was staying a few days earlier at census time. So it seems likely that the lodging house was a place where women in difficult circumstances could go in order to give birth away from the gaze of their family, and that using an assumed name/birthplace would be not uncommon.
One useful piece of information would be to ask your elderly uncle to look at the lower part of the "original" certificate in his possession and discover the date on which the certificate was issued. If that date was at or soon after the date on which the birth was registered, then you can be absolutely certain that you have the correct birth certificate for your grandfather. But if your uncle's certificate was issued at a significantly later date, then there's a question mark over whether it was issued to someone who had guessed that this was the most likely birth and that everything discovered subsequently may be based on supposition.
If that "original" certificate was issued after the time that Tom Mac Gain Harding had left his wife and family (1930ish ?) then you may never be able to be sure about having found the correct birth. Sorry to sound negative, but it's possible that Tom's birth mother gave birth somewhere in Birkenhead on the 4th April 1891 and used an as-yet-untraceable surname to register the birth (possibly her own name). Then perhaps she handed Tom over to an intermediary with a note explaining that he was born on 4th April and that his father was a Mr Mac Gain. Or would it have been necessary for her to have handed the baby over with a birth certificate ?
My genealogy email address is:
faircork-tree (AT) yahoo.co.uk
Jean