A presidential candidate in Honduras backed by President Donald Trump is leading the preliminary count to become the Central American nation’s next leader, scrambling a narrative about the U.S. war on drugs that the Trump administration has pushed in recent months.

The United States has carried out more than 20 military strikes on alleged drug-trafficking boats off the coast from Venezuela this fall in the name of unseating an authoritarian leader who the U.S. says heads up a narco-terrorist operation. Yet Mr. Trump last week made endorsements and pledges in Honduras that seemed to contradict those goals.

Mr. Trump endorsed Nasry “Tito” Asfura of the conservative National Party just days before the election, saying the two could work together to combat drug trafficking. Two days later on

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