Cutting impudence faced by Conservative (4)
I believe the answer is:
clip
'cutting' is the definition.
(I know that clip is a type of cutting)
'impudence faced by conservative' is the wordplay.
'impudence' becomes 'lip' (lip is a kind of impudence).
'faced by' says to put letters next to each other.
'conservative' becomes 'C' (UK political abbreviation).
'lip' put after 'c' is 'CLIP'.
(Other definitions for clip that I've seen before include "Shorten" , "hug" , "rate" , "Extract; cartridge" , "sort of joint" .)