I think the edge of the building on the left of the pic could be the senior boys entrance to the Shaftsbury Boys Club part that was in Thomas Street Have put a street view of where I think the pub was
This picture shows the area just further up from the Borough Hotel, along the eastern extremity of Grange Road opposite the Haymarket.
It was taken around 1967, when all the properties in the area were being prepared for demolition to make way for the new tunnel flyovers.
The buildings on the right are the Army Information Office, F & J Bentley’s pawnbrokers and Clifton Cycles.
Further down on the right, across Jackson Street and along Grange Road (which becomes Grange Street) is the Abacus crane works. This section of roadway is now called Borough Road East.
St Mary’s Church can be seen in the far distance across Chester Street.
The big building in the middle distance is the Wellington Hotel and to its left is the market. The market can be seen because the building that was next door to the Wellington, Sturla’s clothing and drapery store, has already been demolished.