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' says to put letters next to each other (in a down clue, letters appear on others).
'him' becomes 'her' (I can't justify 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" .)