Let me think whether to review cashier (4)
I believe the answer is:
fire
'cashier' is the definition.
(cashier can mean to fire or dismiss a soldier)
'let me think whether to review' is the wordplay.
'let me think' becomes 'er' (I can't explain this - if you can you should give a lot more credence to this answer).
'whether' becomes 'if'.
'to review' is a reversal indicator.
'er'+'if'='erif'
'erif' backwards is 'FIRE'.
Can you help me to learn more?
(Other definitions for fire that I've seen before include "Manually operated device for dealing with flames" , "... or combustion" , "Inspire" , "Sack; discharge (a gun)" , "Blaze - sack" .)