Part of garden where canes are destroyed following request... (9)
I believe the answer is:
pleasance
'part of garden' is the definition.
(I've seen this before)
'canes are destroyed following request' is the wordplay.
'are destroyed' indicates an anagram.
'following' says to put letters next to each other.
'request' becomes 'plea' (plea is a kind of request).
'canes' with letters rearranged gives 'sance'.
'sance' after 'plea' is 'PLEASANCE'.
'where' acts as a link.
(Other definitions for pleasance that I've seen before include "Secluded part of garden" , "secluded spot" , "Sane place (anag.)" , "secret garden" .)