WASHINGTON — John R. Bolton has left the Situation Room, and President Trump is left at the table with a giant set of chips set on hot spots around the world.

In Mr. Trump’s view, the clock is ticking: He needs some big victories between now and the election in November 2020. But he also wants to prove that his idiosyncratic approach to foreign policy — as a series of deals rather than a philosophy of how American hard and soft power is deployed — can produce results that have eluded Washington’s foreign policy establishment for a decade or more.

Here’s a look at six issues on the table.

North Korea

Goal: Total, complete and verifiable denuclearization, including an end to North Korea’s intercontinental ballistic missile program.

Ask Mr. Trump about his negotiations with Kim Jong-un, the North Korean leader, and he will tell you he is already winning: He was the first American president to meet a North Korean leader — three times now — and the first to step, briefly, into North Korean territory. He has gotten back the remains of American soldiers and won a pause, which has lasted nearly two years, in nuclear weapons and intercontinental ballistic missile tests. It all led Mr. Trump to declare on Twitter, after his first meeting with Mr. Kim in Singapore, that North Korea was “no longer a Nuclear Threat.”

The only problem is that the North’s nuclear ability has increased since that meeting, by some estimates significantly. Intelligence estimates indicate that the North’s stockpile of fuel has swelled, and so has its missile arsenal. Short-range missile tests have improved Mr. Kim’s ability to strike American bases in South Korea and Japan with a new generation of weapons intended to avoid missile defenses. And the North hasn’t turned over a list of its weapons, missiles and facilities, which was supposed to be the first step.