Ministry identified by wiggly stripe on headgear (10)
I believe the answer is:
priesthood
'ministry' is the definition.
(thesaurus)
'wiggly stripe on headgear' is the wordplay.
'wiggly' is an anagram indicator.
'on' says to put letters next to each other.
'headgear' becomes 'hood' (hood is a kind of headgear).
'stripe' with letters rearranged gives 'priest'.
'priest'+'hood'='PRIESTHOOD'
'identified by' acts as a link.
(Other definitions for priesthood that I've seen before include "Minister's office" , "office in church" , "future for seminarian" , "cleric's office" , "Ordained order" .)