A bit attached to this private tutor? (5,10)

I believe the answer is:

drill instructor

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'a bit attached to' is the definition.
I don't know anything about this answer so I can't judge whether it can be defined by this definition.

'private tutor?' is the wordplay.
'private' becomes 'drill' (I can't explain this - if you can you should believe this answer much more).
'tutor?' becomes 'instructor' (tutor is a kind of instructor).
'drill'+'instructor'='DRILL INSTRUCTOR'

'this' acts as a link.

Can you help me to learn more?

I've seen this clue in The Sunday Times and The Times.

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