Heard successors give up (4)
I believe the answer is:
cede
'give up' is the definition.
(synonyms)
'heard successors' is the wordplay.
'heard' shows a homophone (sound like).
'successors' becomes 'seed' (both can mean one's offspring).
'seed' is a homophone of 'CEDE'.
(Other definitions for cede that I've seen before include "Deliver" , "Surrender or relinquish possession" , "Surrender or yield" , "Allow" , "Give away" .)