Expressed private resentment (7)
I believe the answer is:
rancour
'resentment' is the definition.
(I know that resentment can be written as rancour)
'expressed private' is the wordplay.
'expressed' indicates a 'sounds like' (homophone) clue.
'private' becomes 'ranker' (ranker can informally mean a military private).
'ranker' sounds like 'RANCOUR'.
(Other definitions for rancour that I've seen before include "Bitter feeling or ill will" , "Deep-seated enmity" , "Malice" , "Bitter feeling or ill-will" , "Deep, bitter hate" .)