Quarrel about publicity that can be boring (7)
I believe the answer is:
bradawl
'boring' is the definition.
(I've seen this before)
'quarrel about publicity' is the wordplay.
'quarrel' becomes 'brawl' (brawling is a kind of quarrelling).
'about' is an insertion indicator.
'publicity' becomes 'ad' (short for advert).
'brawl' enclosing 'ad' is 'BRADAWL'.
'that can be' acts as a link.
(Other definitions for bradawl that I've seen before include "Piercing tool" , "sharp-pointed object" , "Tool for making holes in wood" , "Hole-making tool" , "it has a point" .)