Hard missile in public school (6)
I believe the answer is:
harrow
'public school' is the definition.
(English public school)
'hard missile' is the wordplay.
'hard' becomes 'h' (abbreviation used in pencil classifications).
'missile' becomes 'arrow' (arrow is a kind of missile).
'h'+'arrow'='HARROW'
'in' is the link.
(Other definitions for harrow that I've seen before include "English public school - farm implement" , "Soil cultivator - English public school" , "Agricultural implement - cause pain to (feelings)" , "Soil-breaker; school" , "Clod breaker" .)