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John just came back from New York for a school competition game.

Is the sentence above correct?

Thank you
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Competition game - don't use both words.

This sentence does not mean that the game was in New York. The game is wherever John is now. Is that what you wanted?
Comments  
Teachers: We supply a list of EFL job vacancies
Thank you very much Nona!

John came back from a game in NY.

Is the above sentence correct?
yes.