Many a centre came out of ecumenical order (6)
I believe the answer is:
nuclei
'many a centre' is the definition.
(plural of 'nucleus')
'came out of ecumenical order' is the wordplay.
'out of' suggests deleting specific letters ('out of' can mean no longer having).
'order' is an anagram indicator.
'ecumenical' with 'came' removed is 'uenicl'.
'uenicl' anagrammed gives 'NUCLEI'.
(Other definitions for nuclei that I've seen before include "Atomic cores" , "Central cores" , "Atom cores" , "Centres of atoms" , "Centres of growth or nerve cells - Latin plural" .)