Offender's terrible crimes -- a case of neglect (9)
I believe the answer is:
miscreant
'offender's' is the definition.
(miscreant is a kind of offender)
'terrible crimes a case of neglect' is the wordplay.
'terrible' indicates an anagram.
'case of' suggests removing the centre (outsides of).
'neglect' with its middle taken out is 'nt'.
'crimes' is an anagram of 'miscre'.
'miscre'+'a'+'nt'='MISCREANT'
(Other definitions for miscreant that I've seen before include "behaving unlawfully" , "Wrongdoer" , "culprit" , "Wretch" , "scoundrel" .)