Spot composer dozing, off the record! (6)

I believe the answer is:

measle

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'spot composer dozing off the record' is the definition.
I know nothing about this answer so I can't judge whether this works.

'spot composer dozing off the record' is the wordplay.
'spot composer' becomes 'me' (I can't explain this - if you can you should believe this answer much more).
'dozing' becomes 'asleep' (similar in meaning).
'off' is a deletion indicator ('off' can mean to remove).
'the record' becomes 'EP' (abbreviation for extended play record).
'asleep' with 'ep' removed is 'asle'.
'me'+'asle'='MEASLE'

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I've seen this clue in the Sydney Morning Herald.

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