On radio, outlets for publicising cutting tool (4)
I believe the answer is:
adze
'cutting tool' is the definition.
(adze is a kind of cutting tool)
'on radio outlets for publicising' is the wordplay.
'on radio' indicates a 'sounds like' (homophone) clue.
'outlets for publicising' becomes 'ads' (advertisements).
'ads' is a homophone of 'ADZE'.
(Other definitions for adze that I've seen before include "Something with sharp edge" , "Woodworker's tool" , "Wood-cutting tool" , "Cutting tool with arched blade" , "One might cut" .)