Was busy with a light that needs fixing? (7)

I believe the answer is:

bustled

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'that needs fixing?' is the definition.
I can't tell whether this defines the answer.

'was busy with a light' is the wordplay.
'was busy' becomes 'bust' (I can't explain this - if you can you should give a lot more credence to this answer).
'with' says to put letters next to each other.
'a light' becomes 'led' (abbreviation for light emitting diode).
'bust'+'led'='BUSTLED'

Can you help me to learn more?

I've seen this clue in The Guardian.

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