Resentment of private picked up (7)
I believe the answer is:
rancour
'resentment' is the definition.
(I know that resentment can be written as rancour)
'private picked up' is the wordplay.
'private' becomes 'ranker' (ranker can informally mean a military private).
'picked up' indicates a 'sounds like' (homophone) clue.
'ranker' is a homophone of 'RANCOUR'.
'of' is the link.
(Other definitions for rancour that I've seen before include "Spiteful hostility" , "'Bitterness, resentment (7)'" , "Deep, bitter hate" , "Bitterness and bile" , "Animosity" .)