Conceited runner? Not half! (4)
I believe the answer is:
smug
'conceited' is the definition.
(I've seen this before)
'runner? not half' is the wordplay.
'runner?' becomes 'smuggler' ('smuggler' can be a synonym of 'runner').
'not half' suggests halving the letters.
'smuggler' with half the letters taken is 'SMUG'.
(Other definitions for smug that I've seen before include "Arrogantly complacent" , "Steal" , "Spruce" , "Irritatingly pleased with oneself" , "Be envious of" .)