Big church, not English cathedral, hosting small troupe of musicians (10)
I believe the answer is:
minstrelsy
'troupe of musicians' is the definition.
(I've seen this before)
'big church not english cathedral' is the wordplay.
'big church' becomes 'minster'.
'not' is a deletion indicator.
'english' becomes 'e' (abbreviation).
'cathedral' becomes 'elsy' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'minster' with 'e' taken away is 'minstr'.
'minstr'+'elsy'='MINSTRELSY'
'hosting small' acts as a link.
I am not very happy about this link. Some or all of it may belong to another bit of the clue.
(Other definitions for minstrelsy that I've seen before include "Troupe of musicians" , "musical performance" , "singing" .)