Crossword Genius

Fell asleep, being unable to stay on (7,3)

Ross

I believe the answer is:

dropped off

I'm a little stuck... Click here to teach me more about this clue!

'fell asleep' is the definition.
('drop off' can be a synonym of 'fall asleep')

'being unable to stay on' is the wordplay.
I cannot really understand how this works, but
'being' could be 'dr' (Dr is an example) and 'dr' is found within the answer.
This explanation may well be incorrect...

Can you help me to learn more?

I've seen this clue in The Irish Independent.
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