Meanwhile, in Washington, foreign relations experts noted that with 90 percent of North Korea's exports going through China, Beijing is the one calling the shots.
Pompeo and Wang openly disagreed over which side had called off a two-way security dialogue that had been planned in Beijing this month.
That trade war has overflowed into other areas. USA officials have thus far said the penalties will remain in place until the North's denuclearization is fully verified.
Meanwhile, Trump has accused Beijing of "meddling" in the upcoming mid-terms by running a four-page editorial-style supplement in the Des Moines Register, trying to tap into Iowa farmers' frustration over the tariffs.
Last week, Vice President Mike Pence stepped up the USA pressure campaign against Beijing, going beyond the trade war by accusing China of both "malign" efforts to undermine President Donald Trump ahead of next month's congressional elections and of reckless military action in the South China Sea.
The United States has also angered China with arms sales to Taiwan and new rules allowing top-level U.S. officials to travel to the island, though Washington still recognises Beijing over Taipei. Trump said on Twitter that, "Progress made on Singapore Summit Agreements!"
The speech further inflamed tensions.
Pence also assailed China over its actions in the South China Sea, where the US has deployed warships on tense missions to counter Chinese claims to territory around reclaimed rocks and reefs.
Moon's Europe tour, scheduled for October 13 to 21, is primarily to attend the Asia-Europe Meeting in Belgium and includes stops in France, Italy and Denmark, his office said.
"China and USA relations are at an important juncture and facing challenges", Yang said.
Turkish authorities believe Saudi journalist murdered inside consulate
Mr Khashoggi, 59, has more than 1.6 million Twitter followers and has written for the Washington Post opinion section. Turkish officials told Reuters at the weekend they believed he had been killed inside the Saudi consulate.
Trump emphasised the rapport he says he's developed with the North Korean leader as they negotiate over United States demands that Kim abandon his country's nuclear weapons and ballistic missile programmes. "We will be having more frequent and higher level working level group discussions than we have had in some time".
"When you speak the same language it is because you have the same mother, and this gives us hope", the pope said.
DPRK is the acronym for the North's official name, the Democratic People's Republic of Korea.
Pompeo's visit to Pyongyang marked the fourth of its kind.
The secretary of state said the two sides were "pretty close" to deciding on a time and place for the second summit, declining to provide details.
Speaking to reporters en route to Japan, Mr Pompeo said the plan was to "develop options for both location and timing" for a second summit, although he doubted they would "get it nailed" down mostly due to scheduling and logistics issues.
State Department spokeswoman Heather Nauert said in a statement Kim had invited inspectors to visit the Punggye-ri nuclear test site to confirm it had been irreversibly dismantled.
South Korean President Moon Jae-in said Monday that he believed Kim would soon meet with the leaders of China and Russian Federation "soon" as a "new order is being created on the Korean peninsula".
The archbishop recounted on his return home that when he had told Kim he would tell the Vatican that the two Koreas were moving toward reconciliation and peace, Kim replied with a bow, saying, "Yes, please", Ansa reported. He also urged world leaders to mediate in the crisis and find a diplomatic solution.