ATLANTA—With a little over three weeks to go before elections in close races for U.S. Senate and governor, an escalating fight between Georgia’s Republican secretary of state and a Democratic-leaning voter registration group is moving to court.

Third Sector Development Inc., the parent nonprofit of the New Georgia Project that has been working to register minority voters, has filed suit in Fulton County Superior Court against Secretary of State Brian Kemp and five county election boards, claiming officials have failed to process tens of thousands of voter applications ahead of the Nov. 4 vote. The National Association for the Advancement of Colored People joined the suit, which calls on the court to force the secretary of state and the boards to speed up processing the applications.