Council secretary's exterior show of support (5)
I believe the answer is:
synod
'council' is the definition.
(I know that synod is a type of council)
'secretary's exterior show of support' is the wordplay.
'exterior' means to remove the middle letters.
'show of support' becomes 'nod' (I can't explain this - if you can you should give a lot more credence to this answer).
'secretary' with its centre taken out is 'sy'.
'sy'+'nod'='SYNOD'
Can you help me to learn more?
(Other definitions for synod that I've seen before include "Ecclesiastical gathering" , "Forum for discussing" , "Council of bishops" , "Church council" , "Ecumenical council" .)