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Mr. Grosso still faces an uphill battle against incumbents with plenty of name recognition and a Republican alternative in challenger Mary Brooks Beatty, who could take her share of votes from non-Democrats and has raised more than $33,000 overall — $8,252 during the reported period —and has about $22,500 on hand for her bid.

Mr. Grosso said he remains confident.

“That’s the beauty of this race,” he said, “where you have two votes.”