Crook in US city with last of beer in mug (10)
I believe the answer is:
lawbreaker
'crook' is the definition.
(lawbreaker is a kind of crook)
'us city with last of beer in mug' is the wordplay.
'us city' becomes 'LA' (short for Los Angeles).
'with' becomes 'w' (abbreviation).
'last of' indicates one should take the final letters.
'in' is an insertion indicator.
'mug' becomes 'beaker' (I've seen this before).
The final letter of 'beer' is 'r'.
'r' going inside 'beaker' is 'breaker'.
'la'+'w'+'breaker'='LAWBREAKER'
'in' acts as a link.
(Other definitions for lawbreaker that I've seen before include "Criminal" , "Villain" , "Delinquent" , "Perpetrator of crime" .)