Despite that, Trump hailed the July 6 letter as a "very nice note", saying "Great progress being made".
The letter's date indicates it was sent before Secretary of State Michael Pompeo's visit to Pyongyang last week, in which the USA and North Korea struggled to agree over plans to denuclearize the Korean peninsula.
Pompeo was in Pyongyang from Friday to Saturday to try to flesh out a bare bones denuclearisation deal made during last month's historic summit between Kim and US President Donald Trump.
In the letter Kim describes his June 12 summit with Trump in Singapore, and the resulting joint statement agreed by both sides, as the "start of a meaningful journey".
Kim expresses "invariable trust and confidence" in the president and wishes that "epochal progress" in promoting relations will "bring our next meeting forward".
Kim also agreed then to repatriate remains of US troops who died during the Korean War six decades ago.
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The secretary of state did not meet Kim as he had on his previous two trips to Pyongyang this year, and after his departure, the North's foreign ministry accused the US of making "gangster-like" demands that it unilaterally give up its nuclear weapons.
The Defense Department estimates there are up to 5,300 sets of United States service member remains still somewhere in North Korea.
The Trump administration has insisted that progress is being made.
North Korean officials did not attend a planned meeting with Americans to discuss the return of United States soldiers' remains, officials have said. The president rattled USA allies by demanding that the other NATO nations pay more toward their own defense, although he stopped short of threatening to pull the United States out of the trans-Atlantic alliance.
The other points included a general commitment to the "complete denuclearization" of the divided peninsula, joint efforts to build "a lasting and stable peace regime", and improved bilateral relations. Pyongyang has asked the UN Command Armistice Commission to upgrade the talks to a higher level to include a US general, South Korea's foreign ministry said.
On Sunday, Pompeo seemingly shrugged off the bombastic statement and said the talks with North Korean officials were conducted in "good faith". Despite the secretary's attempt to put a positive spin on the visit, as soon as he was out of the country, North Korea criticized his "unilateral and gangster-like" demands and called the talks "regrettable".
Trump and Kim's friendship comes just a year after the latter was threatened by the former with "fire and fury like the world has never seen" if he continued to develop nuclear bombs.