None, as it happens, refuse church’s conciliatory offer (5,6)
I believe the answer is:
olive branch
'conciliatory offer' is the definition.
'olive branch' can be an answer for 'offer' (olive branch is a kind of offer). I'm not certain of the 'conciliatory' bit.
'none as it happens refuse church's' is the wordplay.
'none' becomes 'o'.
'as it happens' becomes 'live' (as in live television).
'refuse' becomes 'bran' (I've seen this before).
'church' becomes 'ch' (common abbreviation).
'o'+'live'+'bran'+'ch'='OLIVE BRANCH'
Can you help me to learn more?
(Other definitions for olive branch that I've seen before include "conciliatory gesture" , "Offer of reconciliation" , "One offered in peace" , "that will come to a 'sticky' end" , "Peace gesture" .)