In public, once defeated, give up (7)
I believe the answer is:
concede
'give up' is the definition.
(conceding is a kind of giving up)
'in public once defeated' is the wordplay.
'in' indicates a hidden word.
'CONCEDE' can be found hidden inside 'public once defeated'.
(Other definitions for concede that I've seen before include "Acknowledge defeat" , "Surrender, yield" , "Acknowledge - abandon" , "Make concession" , "Admit something is true" .)