A neighbour's key cut out for crime (13)
I believe the answer is:
housebreaking
'crime' is the definition.
(housebreaking is a kind of crime)
'a neighbour's key cut out' is the wordplay.
'cut' means to remove the last letter.
'out' indicates anagramming the letters (out can mean wrong or inaccurate).
'a'+'neighbours'+'key'='aneighbourskey'
'aneighbourskey' with its final letter removed is 'aneighbourske'.
'aneighbourske' is an anagram of 'HOUSEBREAKING'.
'for' acts as a link.
(Other definitions for housebreaking that I've seen before include "Robbery" , "Burglary" , "crime" .)