Play bagging nation's first award for excellence on Broadway (4)

I believe the answer is:

tony

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'award for excellence on broadway' is the definition.
I can't tell whether this definition defines the answer.

'play bagging nation's first' is the wordplay.
'play' becomes 'toy' (synonyms).
'bagging' indicates putting letters inside (I've seen this in other clues).
'nation's first' becomes 'n' (tst letter of 'nation').
'toy' going around 'n' is 'TONY'.

Can you help me to learn more?

(Other definitions for tony that I've seen before include "award in America" , "Boy" , "award for work in theatre" , "- Curtis, US actor" , "Prime Minister of the UK" .)

I've seen this clue in The Guardian.

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