Agree aunt's broken promise (9)
I believe the answer is:
guarantee
'promise' is the definition.
(guaranteeing is a kind of promising)
'agree aunt's broken' is the wordplay.
'broken' indicates anagramming the letters.
'agree'+'aunt'='agreeaunt'
'agreeaunt' with letters rearranged gives 'GUARANTEE'.
(Other definitions for guarantee that I've seen before include "Undertake" , "Formal promise of action" , "Provide financial security for" , "Something ensuring a particular outcome" , "Ensure" .)