SINGAPORE—President Donald Trump launched a high-stakes diplomatic effort to rid North Korea of its nuclear weapons, betting that a friendlier approach with North Korea’s Kim Jong Un—including a surprise promise to halt military exercises with South Korea—would work better than intimidation.

With the task ahead to make the agreement concrete, Secretary of State Mike Pompeo is scheduled to travel to Seoul on Wednesday for talks with South Korean and Japanese officials, followed by a trip to China. The decision to cancel the...