Extremely savvy academic backing church council (5)
I believe the answer is:
synod
'church council' is the definition.
(type of church assembly)
'extremely savvy academic backing' is the wordplay.
'extremely' says to hollow out the word (remove centre letters) (only the extremes of the word are used).
'academic' becomes 'don' (type of academic - especially at Oxbridge).
'backing' says the letters should be written in reverse.
'savvy' with its middle taken out is 'sy'.
'don' reversed gives 'nod'.
'sy'+'nod'='SYNOD'
(Other definitions for synod that I've seen before include "Ecclesiastic council" , "Council of clergy to discuss policy" , "cloth gathered here" , "Forum for discussing" , "Ecumenical council" .)