Did bum shut up about duke? (6)
I believe the answer is:
cadged
'did bum' is the definition.
('cadge' can be a synonym of 'bum')
'shut up about duke?' is the wordplay.
'shut up' becomes 'caged' (I've seen this in another clue).
'about' indicates putting letters inside.
'duke?' becomes 'd' (abbreviation).
'caged' enclosing 'd' is 'CADGED'.
(Other definitions for cadged that I've seen before include "Sponged from another person" , "Bummed" , "Scroughed, went around begging" , "Scrounged, borrowed" , "Scrounged, begged" .)