Stomp lumberingly at first into school (4)
I believe the answer is:
plod
'stomp' is the definition.
(thesaurus)
'lumberingly at first into school' is the wordplay.
'at first' indicates taking the first letters.
'into' is an insertion indicator.
'school' becomes 'pod' (I've seen this before**).
The initial letter of 'lumberingly' is 'l'.
'l' inserted inside 'pod' is 'PLOD'.
(Other definitions for plod that I've seen before include "Shamus, say" , "Walk with heavy steps" , "Stomp" , "Walk heavily (like a policeman?)" , "Walk slowly with a heavy tread" .)