In his speech, broadcast early Tuesday on state-run television, Kim urged the United States to take "corresponding measures" in exchange for denuclearization steps the North Korean dictator has claimed his country has taken since previous year. Sanctions have also been strengthened since the meeting with Mr Trump.
Kim also wants the Trump administration to stop carrying out military exercises with South Korea.
At the time the two leaders signed a vaguely-worded pledge on denuclearisation of the Korean peninsula, but progress has since stalled with Pyongyang and Washington arguing over what that means.
Mr Kim also met with President Moon three times last year - twice at the border truce village of Panmunjom and once in the North's capital Pyongyang - and at the weekend vowed to meet Moon "frequently" this year.
The hermit-kingdom despot used his New Year address to stage the intervention.
It was an occasion the communist leader says he would be keen to repeat: "I am ready to meet with the U.S. president again at any time and will make efforts to surely create outcomes welcomed by the global community", Kim said.
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Trump has been considering a second summit with Kim to speed up the stalling denuclearization.
In the clip, Trump claimed his government had taken steps to improve U.S. relations with North Korea.
Still, that summary did quote Kim as urging North Koreans to effect "a revolutionary surge on all fronts of the socialist construction by regarding self-reliance as a treasured sword for prosperity".
Dressed in a Western style suit, Kim spoke from a dark leather armchair in an office lined with packed bookshelves and paintings of his father, Kim Jong Il, and grandfather Kim Il Sung - a setting that resembled more intimate addresses from the White House by the US president. Taking matters into their own hands, Kim and his South Korean counterpart Moon Jae-in held a symbolic ceremony to link their railway and road networks late December.
Among the topics the leader is expected to address Tuesday are the state - and perhaps future - of denuclearization talks with the US, relations with South Korea and the Hermit Kingdom's economy, which has been tanked by sanctions from the USA and worldwide community after years of nuke tests.
The hardening stalemate has fuelled doubts on whether Kim will ever voluntarily relinquish the nuclear weapons and missiles he may see as his strongest guarantee of survival.
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Kim has said he still stands behind the denuclearization commitment he made to President Trump during the pair's historic Singapore summit, but little progress has been made since June.
The statement jarred with Seoul's claim that Kim is genuinely interested in negotiating away his nukes and suggested that the North will potentially demand the United States withdraw or significantly reduce the 28,500 American troops stationed in South Korea, a major sticking point in any disarmament deal.
However, there was a clear warning that the country is not ready to disarm unilaterally.
Washington and Pyongyang have yet to reschedule a meeting between U.S. Secretary of State Mike Pompeo and senior North Korean officials after the North cancelled it at the last minute in November.
"Kim Jong-un drew from that playbook in his speech today by offering to take actions that might be popular in South Korea but would require USA consent to lift sanctions".
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