Keep an eye on past bishop's responsibility (7)
I believe the answer is:
oversee
'keep an eye on' is the definition.
(to oversee is to watch over)
'past bishop's responsibility' is the wordplay.
'past' becomes 'over' (I've seen this in another clue).
'bishop's responsibility' becomes 'see' (jurisdiction of a bishop, eg the Holy See).
'over'+'see'='OVERSEE'
(Other definitions for oversee that I've seen before include "Watch people toil" , "Be responsible for" , "Supervise, keep an eye on" , "survey" , "take charge" .)