Specifically, Pompeo revealed that North Korea agreed to allow global inspectors to survey their nuclear and missile testing sites.
Trump, tweeting from Washington shortly after Pompeo left North Korea on Sunday, cited progress Pompeo had made on agreements he and Kim committed to at their June meeting in Singapore and said, "I look forward to seeing Chairman Kim again, in the near future".
Since Donald Trump and Kim Jong-un met in Singapore in June, US-North Korea relations have stalled. "We demand that the USA side stop these kinds of mistaken actions".
He earlier said "significant progress" was made during his visit toward an agreement for North Korea to give up its nuclear weapons.
Secretary of State Mike Pompeo arrives at Osan Air Base in Pyeongtaek on Sunday after his Pyongyang trip.
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.
Pompeo declined to comment on the status of talks over Yongbyon or the details of negotiations generally, saying "we're not going to talk about where we are in these negotiations, except for things we have agreed to release with the North Koreans".
Some experts say that opening the Tongchang-ri or Punggye-ri sites for inspection could be a goodwill gesture but has little significance in quickening denuclearisation. "We have to see whether North Korea will let inspectors sample the soil at the Punggye-ri site for detailed analysis, or simply confirm that the site explosion was real".
"These actions have affected the mutual trust between both sides and have cast a shadow over the prospect of China-US relations, which completely go against the interest of our two peoples", Wang said.
In Seoul, Pompeo told South Korean President Moon Jae-in that Kim had agreed to have another summit "as soon as possible", according to a statement from Moon's office.
"But with respect to our relationship with the Republic of Korea and Japan on this issue, I have found that we are in lockstep on the most important issues and how to approach them", Pompeo said.
"The issues that you characterised, we have a fundamental disagreement", Mr Pompeo responded.
KCNA did not mention that issue.
Pompeo even invoked the cancellation of a meeting between US Defense Secretary Jim Mattis and his Chinese counterpart, which was supposed to take place in Beijing this month.
The KCNA said that Kim explained in detail the "proposals" meant to resolve the denuclearization issue and other matters, adding that the leader and Pompeo also exchanged "constructive" views on relevant matters.
"I hope your trip to North Korea and the upcoming second US-North Korea summit will provide a good opportunity for achieving irreversible, decisive progress in terms of denuclearisation and the peace process on the Korean peninsula".
The inspectors will be allowed in as soon as the two sides agree on "logistics", Pompeo told reporters in Seoul before leaving for Beijing on a whirlwind diplomatic trip.