Break that’s rough on ministries (12)
I believe the answer is:
Here is my best explanation:
'break' is the definition.
('intermission' can be a synonym of 'break')
'rough on ministries' is the subsidiary indication.
'rough' is an anagram indicator.
'onministries' with letters rearranged gives 'intermission'.
'that's' is the link.
(Other definitions for intermission that I've seen before include "Interval", "Entr'acte", "Theatre's half-time", "Pause, break" and "Break".)