A motive lacking in each crime (5)
I believe the answer is:
arson
'crime' is the definition.
(crime of illegally starting a fire)
'a motive lacking in each' is the wordplay.
'motive' becomes 'reason' (reason is a kind of motive).
'lacking in' suggests deleting specific letters.
'each' becomes 'ea' (abbreviation).
'reason' with 'ea' taken away is 'rson'.
'a'+'rson'='ARSON'
(Other definitions for arson that I've seen before include "Fire-lighting crime" , "Felonious burning of property" , "Malicious crime" , "burning problem" , "A burning crime" .)