Compete to restrain constant crime (4)
I believe the answer is:
vice
'crime' is the definition.
(vice crime is a kind of crime)
'compete to restrain constant' is the wordplay.
'compete' becomes 'vie' (synonyms).
'to restrain' means one lot of letters goes inside another.
'constant' becomes 'c' (mathematical abbreviation).
'vie' enclosing 'c' is 'VICE'.
(Other definitions for vice that I've seen before include "Fault - gripping tool" , "One that grips" , "Immorality; clamping device" , "Prefix meaning acting as deputy" , "Getting a grip" .)