Influential jazzman has power blowing hard (9)

I believe the answer is:

armstrong

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'influential jazzman' is the definition.
The definition suggests a singular noun which matches the answer.

'power blowing hard' is the wordplay.
'power blowing' becomes 'arm' (I can't justify this - if you can you should believe this answer much more).
'hard' becomes 'strong' (similar in meaning).
'arm'+'strong'='ARMSTRONG'

'has' is the link.

Can you help me to learn more?

(Other definitions for armstrong that I've seen before include "Space pioneer" , "Neil ........., first man on the moon" , "Jazz legend" , "jazz trumpeter" , "jazz musician" .)

I've seen this clue in The Times.

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