Checkmate Broken
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[COMMENT DELETED]
You're missing something, but it probably isn't obvious.
There's a rule called "En Passant" that applies here. Black does have a move available here, and it is hxg3 e.p.
The pawn captures the g pawn "in passing" as if it had only moved one square. This special move is only available if it is played immediately, and only when the opponent's pawn moves forward two spaces on its move to land immediately next to your pawn.