Barnet is after Conservative for seat (5)
I believe the answer is:
chair
'seat' is the definition.
(chair is a kind of seat)
'barnet is after conservative' is the wordplay.
'barnet' becomes 'hair' (I've seen this before).
'is after' means one lot of letters go next to another.
'conservative' becomes 'C' (abbreviation commonly used in UK politics).
'hair' put after 'c' is 'CHAIR'.
'for' is the link.
(Other definitions for chair that I've seen before include "academic office" , "director at meeting" , "Get seat" , "Professoriate" , "Preside, guide meeting" .)