I looked at some electoral registers in the Local Studies Library:
Robert and Priscilla JONES are at 3ct,1 Burns Road in the roll for 1935/6.
In the volume for 1936/7 the voters listed are William Henry and Eva COATES.
So they must have left Burns Road between those dates..
I tried looking for the later address but I was confused by the fact, which I now see, that you have given two different street names 'St Mary Road' which I automatically assumed was St Mary's Road; and then Queen Mary Road, which I'm guessing is the right one.
In 1951 there was a Robert Jones at 471 Hastilar Road South which I think must be in the same area as Queen Mary Road.
edit: I meant to add that there were no electoral rolls in Sheffield after 1939/40 until May 1945.
Thanks Hugh for trying to find information for me.
I am sorry about the address. It should have read Queen Mary Road.
It is funny that the address for Robert Jones in 1951 was 471 Hastilar Road South. That is the address given on my grandfather's second wifes death certificate in 1951.
Another coincidence is that Priscilla was the name of my grandmother.
I think in an earlier directory my grandfather is down as Hugh Dranfield instead of Dronfield.