Trump met with Kim in Singapore in June and - at the time - announced that the two countries are set to "start a new history".
U.S. Secretary of State Mike Pompeo and Chinese Foreign Minister and State Councillor Wang Yi aired their grievances in the open on Monday during a brief visit to Beijing by Washington's top diplomat, amid worsening relations.
The US has said that it will not make such concessions unless the North takes substantive steps to do away with nuclear weapons.
It was Pompeo's fourth trip to Pyongyang and his first visit since July, when he spent the night and left with little to show for his journey.
"While the US side has constantly escalated trade frictions with China, it has also taken actions regarding Taiwan that harm China's core interests", Wang said.
North Korea recently asked for "corresponding" reciprocal measures for the possible dismantlement of its important Yongbyon nuclear complex, a demand experts say might refer to declaring an end to the war or easing sanctions imposed on the country.
One is a missile engine test facility.
Confidence In Trump Lags Behind Russia's Putin, China's Xi In World Poll
Only 38 per cent of French and 39 per cent of Canadians said they had a positive view of the USA , both down from a year ago . Across all countries, an average of 27 percent of respondents said they had confidence in Trump.
Pyongyang in May failed to follow through with its promises to allow inspectors to assess its demolition of Punggye-ri in what has become a roadblock in the country's denuclearization commitment which was reiterated in June after a meeting between Kim and US President Donald Trump in Singapore. The statement did not provide further details.
In Pyongyang on Sunday, Pompeo and Kim met for about 3-1/2 hours, first in a business session and then in a 90-minute luncheon that the North Korean leader hosted at a state guesthouse, according to the pool report from the lone US journalist allowed to accompany Pompeo on his Asia trip.
In an early Monday dispatch, KCNA called the talks "productive and wonderful" and said "mutual stands were fully understood and opinions exchanged".
"Expressing the belief that the dialogue and negotiations between the DPRK and the US based on the deep confidence between the top leaders of the two countries would continue to develop favorably in the future, too, and a good program for the 2nd DPRK-US summit talks would be provided sooner or later, the Supreme Leader said good-bye to the US secretary of State", the KCNA said in English.
But it did not mention any inspection-related issue.
Pompeo next heads to China where he's slated to meet with President Xi Jinping in Beijing.
North Korea's state newspaper, Rodong Sinmun, featured eight photos of the meeting on its front page, including shots of Kim and Pompeo smiling and shaking hands, as well as some with Kim's sister Kim Yo Jong.
Mr Pompeo and North Korean leader Kim Jong-un held talks over the weekend.