Brief Republican over revolting term (5)
I believe the answer is:
remit
'brief' is the definition.
(both can mean someone's responsibilities)
'republican over revolting term' is the wordplay.
'republican' becomes 'r' (abbreviation).
'over' means one lot of letters go next to another (in a down clue, some letters go over others).
'revolting' is a reversal indicator.
'term' becomes 'time' (term can mean a period of time).
'time' backwards is 'emit'.
'r'+'emit'='REMIT'
(Other definitions for remit that I've seen before include "Send; brief" , "Area of responsibility" , "forward" , "Maybe suspend" , "Pay for timer" .)