Buxton is my Grandmother's maiden name which is how I came to have the photograph. She was born in 1918 and had an elder brother Charles born 1900. Her Grandfather and Great Grandfather were also named Charles Buxton. My Gran's brother Charles was a keen photographer and won a prize in a Sheffield Newspaper for a photograph of my Gran buying an icecream at the age of 7 which would've been 1925 ish.
The Charles Buxton who was a barman in the census was my Gran's Granfather.
...
My Aunt told me that someone in the family ran a big hotel in Sheffield. If it came from my Gran's maternal side the name was Fairhurst.
I've attached the other photo which is also a postcard and on the back is printed ; Photo by E. Freeman, 456 Cleethorpes Rd Grimsby.
Hope this helps,
Lisa
Seems odd for a Grimsby photographer to be plying their trade in Sheffield, you would think. Maybe we should look to Grimsby for this hotel?
I think it isalmost certainly the same doorway in the second photograph, Lisa. And you can also see a glimpse of the lower level window, suggesting a basement level for the hotel.
I can find three Charles Buxton's on the 1901 census showing as being born 1900: one is in Stoke Newington, one is the son of a Stone Waller's labourer, living in Nether Hallam, Sheffield, and the last one is the son of a Gas Stoker living in Brightside Bierlow.
Nothing very conclusive from that, I fear.

Paul