He initially fobbed off the question, then said to the reporter: "Are you talking to me?"
Mason said: "Yes, it is just a follow-up of what I just asked you, sir."
Trump replied: "We have the president of Finland. Ask him a question."
"I have one. I just wanted to follow up on the one I asked you," Mason said.
"Did you hear me? Ask him a question. I have given you a long answer. Ask this gentleman a question. Don't be rude," said Trump.
"No sir. I don't want to be rude. I just wanted you to have a chance to answer the question I just asked you," Mason responded.
Mr Trump then made accusations of treason, slamming the "corrupt media" as "truly the enemy of the people".
"I answered everything," Trump said. "It's a whole hoax, and you know who is playing into the hoax, people like you and the fake news media that we have in this country, and I say in many cases, the corrupt news media. Because you are corrupt. Much of the media in this country is not just fake, it's corrupt. And you have some very fine people too. Great journalists. Great reporters. But to a large extent, it is corrupt and it is fake."
READ MORE:
• Secretary of State Mike Pompeo admits he was listening in on Donald Trump's phone call with Ukraine
• Trump's Ukraine scandal 'worse than Watergate'
• The Leighton Smith Podcast 36: Trump, censorship and climate change
• World reacts as transcript of Donald Trump's call with Ukrainian leader is released
At the end of the press conference, Mason said he was just trying to get an answer to his question. He eventually tried to move on to another question, but Trump cut him off, saying: "You never had wins with other presidents, did you? Your wins are now."
Niinistö interjected: "But I think the question is for me?"
During his opening statement, Niinistö told Trump: "Mr President, you have here a great democracy. Keep it going on."
The remark sparked speculation online over whether the Finnish leader was being genuine or sarcastic in the context of his outburst at the journalist:
The only Finnish journalist who got to ask a question asked Mr Niinistö: "What kind of favours has Trump offered you?"
It appeared to be a reference to the Ukraine phone call scandal, but Trump did not address it, saying: "I think you mean the other way around."