WASHINGTON—Primaries in New York, Virginia and Kentucky on Tuesday will test the energy of Democrats’ activist wing, which has fielded candidates promoting a more muscular role for government at a time of economic and social unrest.

Turnout is a wild card in all the contests, given states’ reliance on mail-in absentee voting during the coronavirus pandemic. Kentucky’s voting will be closely watched because Jefferson County, home to Louisville and a large share of the state’s African-American voters, has only one in-person…

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