Writing very quickly? Hey! Song part needs a rewrite! (11)

I believe the answer is:

stenography

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'writing' is the definition.
(stenography is a kind of writing)

'quickly? hey song part needs a rewrite' is the wordplay.
'hey song part' anagrams to 'STENOGRAPHY'.
Though, I am unsure how this is indicated.

'very' is the link.

Can you help me to learn more?

(Other definitions for stenography that I've seen before include "Secretarial skill" , "Shorthand writing" .)

I've seen this clue in The Independent.

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