The Other Mike wrote:
Handsworth is presumably "Sheffield Attercliffe" (1885-2010) but it
doesn't explain the absence of the other constituencies.

That's the puzzle, in the currency of the map Handworth had nowhere
near enough population to be a constituency, and the County Borough
of Sheffield had five constituencies within its boundaries,
Sheffield Attercliffe, Sheffield Brightside, Sheffield Central,
Sheffield Ecclesall and Sheffield Hallam, more-or-less each of
the townships in http://en.wikipedia.org/wiki/File:Sheffield1832.jpg
with Hallam and Ecclesall continuing westwards to the city boundary.

JGH