Tensions on the Korean peninsula have increased recently, with Pyongyang threatening more nuclear and missile tests, which would embarrass Beijing.
This runs directly contrary to Trump's policy of tightening economic sanctions against the North and even threatening military action to force Pyongyang to abandon its nuclear weapons program.
Last week, North Korea's Ministry of State Security, its equivalent to the Central Intelligence Agency, accused the Central Intelligence Agency and NIS of infiltrating the North to commit "state-sponsored terrorism" against Kim Jong-un, trying to assassinate him with a chemical weapon.
On the other hand, Trump shocked the South Koreans by announcing at the end of April that South Korea would have to pay $1 billion for the THAAD system, despite an existing agreement that the United States would bear the cost.
"I'm sure we'll have others join in from across the intelligence community to try and focus this effort so that we can get back on our front foot with respect to foreign intelligence collection against the North Koreans and the capacity, the impact what Kim Jung Un is actually doing", he said.
The US man, Kim Hak-Song, was detained for "hostile acts", the official KCNA news agency said, adding he had worked for the Pyongyang University of Science and Technology (PUST).
The North further said that a South Korean agent named Jo Ki Chol and a "secret agent" named Xu Guanghai, director general of the Qingdao NAZCA Trade Co.
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"According to our law, the Central Public Prosecutor's Office of the DPRK will use all available methods to start to work to demand the handover of the criminals involved, so as to punish the organizers, conspirators and followers of this awful state-sponsored terrorism", Han said in the extradition statement.
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Earlier in May, the North Korean Ministry of State Security alleged that the CIA and South's National Intelligence Service (NIS) had hired an assassin named only as Kim to kill their leader.
Their report said Mr Han had "declared the principled stand of the North Korean government to find out all of the terrorist maniacs and mercilessly wipe them out".
North Korea held a military parade, rolled out a few missile mock-ups and tested a missile, which blew up nearly immediately after launch.
South Korean protesters burn Kim Jong-un's placards during an anti-North Korea rally.
Beijing announced last month that it would continue to stage live fire drills and test new weapons to address national security threats, as a response to the US' recent controversial deployment of its THAAD anti-missile system in South Korea.
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