Push skiver, finally, into school (4)
I believe the answer is:
prod
'push' is the definition.
(prodding is a kind of pushing)
'skiver finally into school' is the wordplay.
'finally' suggests the final letters.
'into' means one lot of letters goes inside another.
'school' becomes 'pod' (I've seen this before**).
The last letter of 'skiver' is 'r'.
'r' put into 'pod' is 'PROD'.
(Other definitions for prod that I've seen before include "Poke, jab, nudge" , "Dig" , "Poke with forefinger, stimulate" , "Poke, stimulate into action" , "Give a reminder" .)