SEOUL—When Secretary of State Mike Pompeoarrived in Pyongyang this month to discuss North Korea’s nuclear program, leader Kim Jong Un was out of town—visiting a potato farm and factory in the country’s impoverished north, according to state media.

Mr. Kim, having declared his nuclear program complete and claimed his place on the world stage next to the U.S. president at a Singapore summit, has begun to lavish attention on the livelihood of his people.

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