Dad's not heavy on him! (6)
I believe the answer is:
father
'dad's' is the definition.
(I've seen this before)
'heavy on him' is the wordplay.
'heavy' becomes 'fat' (similar in meaning).
'on' is a charade indicator (letters next to each other) (in a down clue, letters appear on others).
'him' becomes 'her' (I can't explain this - if you can you should give a lot more credence to this answer).
'fat'+'her'='FATHER'
'not' is the link.
Can you help me to learn more?
(Other definitions for father that I've seen before include "Sire" , "pops" , "relation" , "and 13: Jolly gift-deliverer" , "Relative and priest" .)