Award finally stops shy private becoming depressed (4,4)
I believe the answer is:
cast down
'depressed' is the definition.
(cast down can mean depressed or downbeat)
'award finally stops shy private' is the wordplay.
'finally' suggests the final letters.
'stops' indicates putting letters inside (stopping can mean blocking a gap).
'shy' becomes 'cast' (both can mean to throw).
'private' becomes 'own' (as in one's own or private belongings).
The final letter of 'award' is 'd'.
'cast'+'own'='castown'
'd' inserted into 'castown' is 'CAST DOWN'.
'becoming' is the link.
(Other definitions for cast down that I've seen before include "Gloomy" , "Depressed" , "disheartened" , "Dishearten" , "Discouraged" .)