Despatched the aroma, I hear (4)
I believe the answer is:
sent
'despatched' is the definition.
(to despatch is to send off)
'aroma i hear' is the wordplay.
'aroma' becomes 'scent' (synonyms).
'i hear' shows a homophone (sound like).
'scent' is a homophone of 'SENT'.
'the' acts as a link.
(Other definitions for sent that I've seen before include "charged with emotion?" , "Posted" , "Dispatched" , "old money in Tallinn" , "Thrilled" .)