Vice Chairman Kim Yong Chol noted Saturday that "the world is paying close attention to our meeting".
There had also been worrying signs even before the Pompeo trip.
It was unclear whether Pompeo would meet Kim Jong Un, as he did on his previous trips, before heading for Tokyo later on Saturday.
The alleged gift came after the song became a point of discussion for Trump and Kim during their summit in Singapore last month.
He noted, however, that "there are things that I have to clarify", Reuters reported.
Evans Revere, a former USA diplomat with a long history of negotiating with North Korea, said that it was evident now that talks in Pyongyang had not gone well - and that it appeared North Korea may have no intention of actually denuclearizing in the way the United States would want. "It's a very, very unrealistic plan", Sue Mi Terry, Korea an analyst for the Center for Strategic and International Studies and former Central Intelligence Agency analyst, told Morning Edition on Friday, adding that just the process of verifying the full scope of North Korea's nuclear program "takes years, decades".
Pompeo himself had been asked about the CD earlier by a journalist accompanying him to North Korea.
North Korea said Saturday that high-level talks with a US delegation led by Secretary of State Mike Pompeo were "regrettable" and accused Washington of trying to unilaterally pressure the country into abandoning its nukes.
According to the spokesman, during the talks with Pompeo, North Korea raised the issue of a possible declaration to formally end the 1950-53 Korean war, which concluded with an armistice and not a peace treaty.
While Pompeo hailed progress at the latest talks, the KCNA commentary and the results he announced on Saturday will only add to unease that North Korea's commitment to denuclearisation is a facade, and it has no intention of giving up its weapons.
Pyongyang said the USA had betrayed the spirit of the summit by making "one-sided and robber-like" demands on the denuclearisation of North Korea.
Trump has declared on Twitter that North Korea no longer poses a nuclear threat.
CVID stands for complete, verifiable and irreversible dismantlement of the North's nuclear program, an oft-cited demand by Washington for Pyongyang.
That change raised suspicion that the U.S. was softening its demands for the country, an argument that State Department spokeswoman Heather Nauert had insisted on Friday wasn't true.
Kim Jong-un has promised to work towards denuclearisation, but details on how this will be achieved remain thin.
But fissures have emerged in the U.S. stance.
Speaking privately, USA officials suggested the North Korean statement was a negotiating tactic.
Pompeo also told reporters that he discussed the destruction of a missile test site that Trump said was already being dismantled. Nauert later said the United States wasn't putting a timeline on the process.
Heather Nauert, the State Department spokesperson, tweeted a number of photos during the talks, including one of a beaming Mr Pompeo pictured after the final meeting.
The talks were held at a villa in an official compound close to the imposing mausoleum where North Korea's former helmsmen Kim Il-sung and Kim Jong-il - the current leader's grandfather and father - lie in state.
The contrast between Pyongyang and Washington cast a cloud over future negotiations, raising questions over whether the North is committed to abandoning the nuclear programmes it has developed for decades and sees as key to its survival.