Applaud principal head wanting increment (5)
I believe the answer is:
raise
'increment' is the definition.
(raise is a kind of increment)
I don't understand the rest of the clue.
Can you help me to learn more?
(Other definitions for raise that I've seen before include "Increased earnings in America" , "jack up" , "Lift; (to) levy (money)" , "Inflate" , "Move higher" .)