My being under doctor goes on (6)
I believe the answer is:
drones
'goes on' is the definition.
'my being under doctor' is the wordplay.
'my' becomes 'ones' ('one's' is a humorous equivalent to 'my').
'being under' means one lot of letters go next to another (in a down clue, some letters go below others).
'doctor' becomes 'dr' ('Dr' can be a synonym of 'doctor').
'ones' after 'dr' is 'DRONES'.
(Other definitions for drones that I've seen before include "Stingless male bees" , "Male honey-bees" , "Honeybees that mate with the queen" , "Bass pipes of bagpipes" , "Male honey bees" .)