I don't have directories that go that far but I know someone who does so I will ask him. Are you sure they moved from Burns Road when Gerald went to Western Road School as it's not far away? Which years do you believed they moved?
Looking at the school entries on this site Gerald, my uncle, was at Burns Road until 1929. Then records show that my mum was still at Burns Road in 1932. Reason for leaving was to move on to secondary.
The year on the certificate of Merit is 1934 but that might just be the year that kind of certificate came in to being.
They are definitely in Queen Mary Road in 1942 when my grandmother died.
If you can access directories for the period 1934 to 1942 that would be great. Even if it just tells us how long prior to 1942 they were in Queen Mary Road.
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.