Matter hard to press forward (4)
I believe the answer is:
push
'press forward' is the definition.
(I've seen this before)
'matter hard' is the wordplay.
'matter' becomes 'pus' (pus is a kind of matter).
'hard' becomes 'h' (abbreviation used in pencil classifications).
'pus'+'h'='PUSH'
'to' is the link.
(Other definitions for push that I've seen before include "Promote - drive" , "Drive forward" , "Try to sell" , "Offensive - press" , "Encourage" .)