Private soldier expressed bitterness (7)
I believe the answer is:
rancour
'bitterness' is the definition.
(synonyms)
'private soldier expressed' is the wordplay.
'private soldier' becomes 'ranker' (I've seen this before).
'expressed' indicates a 'sounds like' (homophone) clue.
'ranker' is a homophone of 'RANCOUR'.
(Other definitions for rancour that I've seen before include "Bitter feeling or ill-will" , "Spite" , "Deep, bitter hate" , "Deep-seated bitterness" , "Bitter resentment" .)