The director's handling of the play performed in The Theatre (9)
I believe the answer is:
treatment
'the director's handling' is the definition.
'treatment' can be an answer for 'handling' ('treatment' can be a synonym of 'handling'). I'm not sure about the 'the director's' bit.
'the play performed in the theatre' is the wordplay.
I cannot really see how this works, but
'theatre' could be 'nt' (National Theatre) and 'nt' is found in the answer.
This may be the basis of clue (or it may be nonsense).
'of' acts as a link.
Can you help me to learn more?
(Other definitions for treatment that I've seen before include "Maintenance work" , "Cure, remedy" , "Medication" , "Management of something, illness say" , "NHS care?" .)