Some New York City voters went to the polls Tuesday to elect a new member of Congress to succeed former U.S. Rep. Michael Grimm, who resigned in January after pleading guilty to federal tax-fraud charges.

The contest between Staten Island District Attorney Daniel Donovan, a Republican, and City Councilman Vincent Gentile, a Democrat, had attracted little national attention despite Mr. Donovan’s role in overseeing the grand jury that declined to indict a police officer in the death of Eric Garner.

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