After Seoul, Pompeo ends his trip Monday in China, North Korea's political and economic lifeline.
He earlier said "significant progress" was made during his visit toward an agreement for North Korea to give up its nuclear weapons.
Wang Yi said he told Pompeo during a meeting Monday that a shadow has been cast "over the future of China-U.S. relations".
This time, Kim was satisfied with the "productive and wonderful talks" with Pompeo, according to the regime's Korean Central News Agency.
Vatican Secretary for Relations with States Paul Gallagher visited the Joint Security Area on the Demilitarized Zone between North and South Korea July 5, where he said, "It is a very historic period, a period of hope, and the Holy Father is supporting that movement".
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The Pope has long urged the two Koreas to make peace.
But, as Pompeo made his way back to Washington after three days of talks in Tokyo, Pyongyang, Seoul and Beijing, it was unclear what had been accomplished.
Chinese State Councilor and Foreign Minister Wang Yi shows the way to U.S. Secretary of State Mike Pompeo before a meeting at the Diaoyutai State Guesthouse in Beijing, China October 8, 2018. That site was dismantled in May before a crowd of journalists, not nuclear experts. "But it is wholly in keeping with North Korea's desire to maintain the upper hand in the global public narrative, shaping a discourse where it is the driver of peace". But what is known appears marginal.
South Korean President Moon Jae-in will deliver Kim's invitation when he meets Francis next week during a trip to Europe, Blue House spokesperson Kim Eui-kyeom said on Tuesday. Trump said on Twitter that, "Progress made on Singapore Summit Agreements!" On that occasion, he did not meet with Kim.
Pompeo declined to comment whether there was progress on a shutdown of the Yongbyon site.
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But post-summit nuclear negotiations between North Korea and the United States got off to a rocky start, with the North accusing Washington of making "gangster-like" unilateral demands for denuclearization, and calling for sanctions to be lifted before any further progress in nuclear talks.
Pompeo is on a whistle-stop tour around Asia mainly focused on North Korea's nuclear program, but upon arrival in Beijing, the agenda swiftly pivoted to the burgeoning acrimony between the world's two biggest economies, sparked by PresidentDonald Trump's trade war and fueled by his allegations that the Chinese government is "meddling" in next month's midterm elections.
During the most recent summit between Korean leaders in September, Kim told Moon that he would "greatly welcome" the Pope to Pyongyang, according to South Korea's presidential office.
Pompeo told South Korean President Moon Jae-in on Sunday his latest trip to Pyongyang was "another step forward" to denuclearisation, but there were "many steps along the way".
"The past four decades have witnessed tremendous changes in both countries and substantial benefits from mutual exchanges", Wang said.
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The pope has yet to respond to Kim's invitation.