Responsibility of working with the Independent (4)
I believe the answer is:
onus
'responsibility' is the definition.
(onus is a duty or responsibility)
'working with the independent' is the wordplay.
'working' becomes 'on' (on can mean working or activated).
'with' means one lot of letters go next to another.
'the independent' becomes 'us' (the puzzle setter and their Independent colleagues).
'on'+'us'='ONUS'
'of' is the link.
(Other definitions for onus that I've seen before include "Burden, duty" , "Obligation - charge" , "Responsibilty" , "Liability" , "'Burden, load (4)'" .)