On Sunday, the North's state media reported that Kim received a personal message "of excellent content" from the U.S. president, just days after Trump said the North Korean leader had sent him a "beautiful letter".
The White House extended last week six executive orders containing sanctions imposed over North Korea's nuclear and missile programmes by one year.
The North Korean Foreign Ministry comments about Mr. Pompeo on Wednesday, meanwhile, came hours before South Korean President Moon Jae-in claimed North Korean and USA officials are holding "behind-the-scenes talks".
Denuclearization talks between North Korea and the U.S. had appeared stalled since the last summit between Trump and Kim ended abruptly in Hanoi without a deal in February.
North Korea criticized the U.S. State Department on Wednesday for issuing human rights reports and targeted Secretary of State Mike Pompeo for supporting sanctions against the regime.
"Regardless of how hard the leaders of the two sides try to establish a new relationship, it would be difficult to expect improvements, or the denuclearization of the Korean Peninsula, if policy makers who harbor hostile sentiments control US politics", the North Korean representative said.
Trump later said that he had written Kim a "very friendly letter" in response to a birthday greeting he had received from him the previous week.
He said the U.S.is prepared to take commitments it made with North Korea at the Singapore summit previous year "simultaneously and in parallel" fashion.More news: Video shows abandoned newborn girl discovered inside plastic bag
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"All these speak clearly to the fact that the wild dream of the United States to bring us to our knees by means of sanctions and pressure has not changed at all but grows even more undisguised", he added.
The spokesman described the renewal of the U.S. bans as a "manifestation of the most extreme hostile acts", but said North Korea "would not thirst for a lifting of [the] sanctions".
Should DPRK-U.S. nuclear negotiations continue in earnest, Moon said, the "key to the negotiations will be to determine what kind of measures that the North will have to complete to say that substantive denuclearization has been achieved".
Trump wanted to travel to the DMZ on a 2017 visit to South Korea but heavy fog prevented it.
"Our state is not a country that will surrender to the USA sanctions, nor are we a country which the USA could attack whenever it desires to do so".
He immediately corrected himself, saying he had misspoken and the figure applied to the effect of U.S. sanctions on Iran, but the remarks were nonetheless deemed "reckless" by Pyongyang. -North Korea talks, Moon said improved economic relations between the Koreas would be "conductive" to the larger nuclear negotiations.
"It will be linked to the definition of denuclearization being clarified, upon which an agreement was not reached at the Hanoi summit", he said.