Enters the exam, exhausted, to get prize (6)
I believe the answer is:
esteem
'prize' is the definition.
(I know that prize can be written as esteem)
'enters the exam exhausted' is the wordplay.
'enters' becomes 'este' (I can't explain this - if you can you should give a lot more credence to this answer).
'exhausted' suggests removing the centre (the word is emptied out or exhausted).
'exam' with its centre taken out is 'em'.
'este'+'em'='ESTEEM'
'to get' is the link.
(Other definitions for esteem that I've seen before include "Favourable regard" , "Set a high value on" , "Revere(nce)" , "Think well of" , "Rate highly" .)