Publicise search for firearm (3,5)
I believe the answer is:
air rifle
'firearm' is the definition.
(I've seen this before)
'publicise search' is the wordplay.
'publicise' becomes 'air' ('air' can be a synonym of 'publicise').
'search' becomes 'rifle' (I've seen this before).
'air'+'rifle'='AIR-RIFLE'
'for' is the link.
(Other definitions for air rifle that I've seen before include "Rail fire (anag) - weapon" , "Shooter" , "Pneumatic arm" , "Pellet gun" , "Pellet firer" .)