CLEVELAND — A battle over early voting hours in Ohio is flaring again after a top adviser to Gov. John Kasich, a Republican, this week made remarks that Democrats cast as racist, and the Republican secretary of state suspended two local election officials who voted to extend balloting hours in one county.

Anger over rules on early voting in this presidential battleground state appeared as if it might ease last week when, under pressure from voters’ rights groups, the secretary of state announced that all Ohio counties would follow a uniform policy over the five-week early voting period that begins Oct 2.

But tensions have done anything but cool. The new policy excluded weekends, and Democrats have accused the secretary of state, Jon Husted, of trying to scale back voting opportunities in urban areas that had longer voting hours during the last presidential election, when Barack Obama won the state. Before Mr. Husted issued his directive, the state’s 88 county election boards, each made up of two Republicans and two Democrats, issued their own rules, and weekend and evening hours varied by county. The new policy allows voting from 8 a.m. to 5 p.m. on weekdays during the first three weeks of the period, and 8 a.m. to 7 p.m. during the final two weeks.

The friction detonated this week when Doug Preisse, the influential Republican Party chairman of Franklin County, which includes the state capital, Columbus, was quoted in The Columbus Dispatch newspaper as saying, “I guess I really actually feel we shouldn’t contort the voting process to accommodate the urban — read African-American — voter turnout machine.”