Son, given heavy responsibility, slept (9)
I believe the answer is:
Here is my best explanation:
'slept' is the definition.
(this has been seen before in a quick crossword)
'son given heavy responsibility' is the subsidiary indication.
'son' becomes 's'.
'given heavy responsibility' becomes 'lumbered'.
(Another definition for slumbered that I've seen is "Slept".)