He may give the mayor a bit of a turn (3)
I believe the answer is:
roy
'he' is the definition.
(I know that Roy is a male given name)
'the mayor a bit of a turn' is the wordplay.
'bit of' indicates a hidden word.
'a turn' is a reversal indicator.
'yor' can be found hidden inside 'mayor a'.
'yor' backwards is 'ROY'.
'may give' acts as a link.
(Other definitions for roy that I've seen before include "- Orbison, US singer" , "-- Hattersley; -- Jenkins" , "Man" , "Rob . . . was a notorious Scottish outlaw" , "Rob --, Scots outlaw" .)