Chore holds up gym assistant (6)
I believe the answer is:
deputy
'assistant' is the definition.
(both can mean a second-in-command)
'chore holds up gym' is the wordplay.
'chore' becomes 'duty' (I've seen this in another clue).
'holds' is an insertion indicator.
'up' says the letters should be written backwards (in down clue: letters go upwards).
'gym' becomes 'pe' (short for Physical Education).
'pe' in reverse letter order is 'ep'.
'duty' placed around 'ep' is 'DEPUTY'.
(Other definitions for deputy that I've seen before include "Subordinate" , "mine safety official" , "One who stands in" , "Stand-in, delegate" , "US sheriff's assistant" .)