A ceremony run without bishop or judge (7)
I believe the answer is:
arbiter
'judge' is the definition.
(thesaurus)
'a ceremony run without bishop' is the wordplay.
'ceremony' becomes 'rite' (rite is a kind of ceremony).
'run' becomes 'r' (cricket abbreviation).
'without' is an insertion indicator ('without' can be similar in meaning to 'outside').
'bishop' becomes 'b' (abbreviation used in chess).
'rite'+'r'='riter'
'riter' enclosing 'b' is 'rbiter'.
'a'+'rbiter'='ARBITER'
'or' is the link.
(Other definitions for arbiter that I've seen before include "Judge to decide a disputed issue" , "One appointed to settle a dispute" , "One decides in disputes" , "Someone with the power to settle the matter" , "One judges a disputed issue" .)