Wanting independence, instructs naughty pets in parts of church (9)
I believe the answer is:
transepts
'wanting independence instructs' is the definition.
I can't judge whether this definition defines the answer.
'naughty pets in parts of church' is the wordplay.
'naughty' indicates anagramming the letters.
'in' is an insertion indicator.
'parts of church' becomes 'trans' (I can't justify this - if you can you should believe this answer much more).
'pets' with letters rearranged gives 'sept'.
'sept' inserted inside 'trans' is 'TRANSEPTS'.
Can you help me to learn more?
(Other definitions for transepts that I've seen before include "In cross-shaped church, these are at right angles to nave" , "parts of cathedrals" , "Sections of church" .)