He could kill staff supporting strike (6)
I believe the answer is:
hitman
'he could kill' is the definition.
(a hitman kills people)
'staff supporting strike' is the wordplay.
'staff' becomes 'man' (as in staffing or manning a particular place).
'supporting' means one lot of letters go next to another (in a down clue, some letters go below others).
'strike' becomes 'hit' (synonyms).
'man' put after 'hit' is 'HITMAN'.
(Other definitions for hitman that I've seen before include "killer" , "Hired gun" , "Assassin" , "offer?" .)