British shunning extra responsibility (4)
I believe the answer is:
onus
'responsibility' is the definition.
(thesaurus)
'british shunning extra' is the wordplay.
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'shunning' indicates named letters should be taken away.
'extra' becomes 'bonus' (I've seen this before).
'bonus' with 'b' taken away is 'ONUS'.
(Other definitions for onus that I've seen before include "Obligation - charge" , "Encumbrance" , "Burden, duty" , "'Responsibility, task or burden (4)'" , "albatross?" .)