WASHINGTON — The White House has rejected a Vladimir Putin-backed effort to hold a referendum in eastern Ukraine on the region's future, distancing itself from the idea in the aftermath of President Donald Trump's controversial summit with the Russian leader.

Russia's ambassador to the U.S., Anatoly Antonov, said the two leaders had discussed the possibility of a referendum in separatist-leaning eastern Ukraine during their Helsinki summit.

But Trump's National Security Council spokesman Garrett Marquis said Friday that agreements between Russia and the Ukrainian government for resolving the conflict in the Donbas region do not include any such option and any effort to organize a "so-called referendum" would have "no legitimacy."

The back-and-forth came as the White House outlined the agenda for a proposed second summit between Trump and Putin — in Washington this fall — that would focus on national security. Moscow signaled its openness to a second formal meeting between the two leaders as criticism of Trump over his first major session with his Russian counterpart kept up in the U.S.

Trump left the White House for his New Jersey golf club for the weekend. Once he got there, he returned to Twitter to complain about news coverage of Monday's meeting.

"I got severely criticized by the Fake News Media for being too nice to President Putin," he tweeted. "In the Old Days they would call it Diplomacy. If I was loud & vicious, I would have been criticized for being too tough."

A White House official said the next Trump-Putin meeting would address national security concerns they discussed in Helsinki, including Russian meddling. The official did not specify if that meant Russia's interference in U.S. elections.

The official, who spoke on condition of anonymity to discuss planning, said the talks would also cover nuclear proliferation, North Korea, Iran and Syria.

One stop Putin almost surely won't make is Capitol Hill.

House Democratic leader Nancy Pelosi urged House Speaker Paul Ryan to make clear that Putin wouldn't be invited to address Congress if he visits Washington.

She said Trump's "frightened fawning over Putin is an embarrassment and a grave threat to our democracy."

Secretary of State Mike Pompeo had a sunnier view of the likely second get-together.

He said at the United Nations he was "happy that the two leaders of two very important countries are continuing to meet. If that meeting takes place in Washington, I think it is all to the good. Those conversations are incredibly important."

It was not clear whether such a meeting would take place before or after the November congressional elections in the U.S.

A White House meeting would be a dramatic extension of legitimacy to the Russian leader, who has long been isolated by the West for activities in Ukraine, Syria and beyond and is believed to have interfered in the 2016 presidential election that sent Trump to the presidency. No Russian leader has visited the White House in nearly a decade.

U.S. officials have been mum on what, if anything, the two leaders agreed to in Helsinki during a two-hour, one-on-one meeting. Director of National Intelligence Dan Coats said Thursday he had yet to be briefed on the private session.

The Russian government has been somewhat more forthcoming.

"This issue (of a referendum) was discussed," Antonov said, adding that Putin made "concrete proposals" to Trump on solutions for the four-year, Russian-backed insurgency in eastern Ukraine, which has killed more than 10,000 people. He did not elaborate on what Putin's solutions would be.

The move may be seen as an effort to sidestep European peace efforts for Ukraine and increase the pressure on the Ukrainian government in its protracted conflict with pro-Russian separatists in the Donbas region.

In a sign of support for the Ukraine government, the Pentagon said Friday it would provide $200 million in additional training, equipment and advisory assistance to Ukraine's military.

Trump tweeted Thursday that he looked forward a "second meeting" with Putin and defended his performance at Monday's summit, in which the two leaders conferred on a range of issues including terrorism, Israeli security, nuclear proliferation and North Korea.

"There are many answers, some easy and some hard, to these problems ... but they can ALL be solved!" Trump tweeted.

In Moscow, Antonov said it is important to "deal with the results" of the first summit before jumping too fast into a new one. But he said, "Russia was always open to such proposals."

The White House is still trying to clean up post-summit Trump statements on Russian interference in the 2016 election. Trump's public doubting of Russia's responsibility in a joint news conference with Putin on Monday provoked withering criticism from Republicans as well as Democrats and forced the president to make a rare public admission of error.

Then on Thursday, the White House said Trump "disagrees" with Putin's offer to allow U.S. questioning of 12 Russians who have been indicted for election interference in exchange for Russian interviews with the former U.S. ambassador to Russia and other Americans the Kremlin accuses of unspecified crimes. Trump initially had described the idea as an "incredible offer."

The White House backtrack came just before the Senate voted overwhelmingly against the idea.