Break that’s rough on ministries (12)
I believe the answer is:
intermission
'break' is the definition.
(I've seen this before)
'rough on ministries' is the wordplay.
'rough' indicates an anagram.
'on'+'ministries'='onministries'
'onministries' with letters rearranged gives 'INTERMISSION'.
'that's' is the link.
(Other definitions for intermission that I've seen before include "Entr'acte" , "Break; respite" , "Theatre's half-time" , "Pause, break" , "Interval" .)