WASHINGTON—Entering a contentious stage in the presidential campaign, President Trump is seeking to focus voters on an economic rebound and law and order, while Democratic challenger Joe Biden is redoubling efforts to make the election a referendum on the president’s performance on the coronavirus pandemic and the economy.

With the start of early voting in some states this month, Mr. Trump and Mr. Biden are fighting for control of a battleground map stretching across a dozen states against the backdrop of a pandemic that…