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American cinema club puts woman first (9)

Ross

I believe the answer is:

hollywood

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'woman first' is the definition.
Although both the answer and definition are singular nouns, I don't understand how they can define each other.

'american cinema club' is the wordplay.
'american cinema' becomes 'holly' (I am not sure about this - if you are sure you should believe this answer much more).
'club' becomes 'wood' (wood is a kind of club).
'holly'+'wood'='HOLLYWOOD'

'puts' acts as a link.

Can you help me to learn more?

(Other definitions for hollywood that I've seen before include "Tinseltown" , "in LA?" , "Where Frankie goes" , "There's a place for tinsel" , "California suburb" .)

I've seen this clue in The Daily Express.
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