Short piece of film has Conservative on edge (4)
I believe the answer is:
clip
'short piece of film' is the definition.
(I've seen this before)
'conservative on edge' is the wordplay.
'conservative' becomes 'C' (UK political abbreviation).
'on edge' becomes 'lip' (I've seen this before).
'c'+'lip'='CLIP'
'has' is the link.
(Other definitions for clip that I've seen before include "Cut short; cartridge holder" , "Fastener; cut short" , "Prune" , "Trim the catridge container?" , "It holds items together" .)