Offends underworld boss with emotional outbursts (8)
I believe the answer is:
disgusts
'offends' is the definition.
(disgust can mean to offend or revolt)
'underworld boss with emotional outbursts' is the wordplay.
'underworld boss' becomes 'dis' (Roman god of the underworld).
'with' means one lot of letters go next to another.
'emotional outbursts' becomes 'gusts' (gust can mean an outpouring of emotion).
'dis'+'gusts'='DISGUSTS'
(Other definitions for disgusts that I've seen before include "Revolts, nauseates" , "nauseating" .)