Heart broken by 'X', bully (8)
I believe the answer is:
threaten
'bully' is the definition.
(thesaurus)
'heart broken by x' is the wordplay.
'broken' is an anagram indicator.
'by' means one lot of letters go next to another.
'x' becomes 'ten' ('ten' can be a synonym of 'x').
'heart' with letters rearranged gives 'threa'.
'threa'+'ten'='THREATEN'
(Other definitions for threaten that I've seen before include "Foreshadow" , "Warn of violence" , "Utter menaces" , "Look likely to" , "Give warning of something unpleasant" .)