Disparager’s bad name sons endure (10)
I believe the answer is:
mudslinger
'disparager's' is the definition.
(mudslinger is a kind of disparager)
'bad name sons endure' is the wordplay.
'bad name' becomes 'mud' (if someone's 'name is mud' they are a disgrace).
'sons' becomes 's'.
'endure' becomes 'linger'.
'mud'+'s'+'linger'='MUDSLINGER'
(Other definitions for mudslinger that I've seen before include "Scandalmonger" , "One making accusations" , "knocker" , "Person trying to damage another's reputation" , "Slanderer" .)