Relation seen in public performance (9)
I believe the answer is:
operation
'performance' is the definition.
(I've seen this before)
'relation seen in public' is the wordplay.
'relation' becomes 'ratio' (ratio is a kind of relation**).
'seen in' is an insertion indicator.
'public' becomes 'open' ('open' can be similar in meaning to 'public').
'ratio' going inside 'open' is 'OPERATION'.
(Other definitions for operation that I've seen before include "Handling" , "performance" , "Theatre procedure" , "working" , "medical treatment" .)