It's Francis's responsibility - after half an hour, start to look with fresh eyes (4,3)
I believe the answer is:
holy see
'it's francis's responsibility' is the definition.
(jurisdiction of the Pope, Pope Francis)
'half an hour start to look with fresh eyes' is the wordplay.
'half' means to take half the letters.
'start to' indicates taking the first letters.
'with' says to put letters next to each other.
'fresh' indicates an anagram.
The first letter of 'look' is 'l'.
'hour' halved is 'ho'.
'eyes' is an anagram of 'ysee'.
'ho'+'l'+'ysee'='HOLY SEE'
'after' is the link.
(Other definitions for holy see that I've seen before include "Church court" , "Sovereign state within Rome" , "The Vatican" , "Pope's position" , "Rome ecclesiastically" .)