On the radio, party sounded relieved (6)
I believe the answer is:
sighed
'sounded relieved' is the definition.
(I've seen this in another clue)
'on the radio party' is the wordplay.
'on the radio' shows a homophone (sound like).
'party' becomes 'side' (eg a side or party in a contract).
'side' is a homophone of 'SIGHED'.
(Other definitions for sighed that I've seen before include "Sounded weary" , "Exhaled softly and sadly" , "Breathed wearily" , "Expressed wordless sorrow" , "Showed exasperation" .)