Student in wager to sing loudly (4)
I believe the answer is:
belt
'sing loudly' is the definition.
(synonyms)
'student in wager' is the wordplay.
'student' becomes 'L' (as in L-plates for learner drivers).
'in' indicates putting letters inside.
'wager' becomes 'bet' (synonyms).
'l' going inside 'bet' is 'BELT'.
'to' acts as a link.
(Other definitions for belt that I've seen before include "Rush, dash" , "Hit - trousers' support" , "A vigorous blow" , "Heavy blow" , "Flog -- band" .)