Clinton called the North Korea test "outrageous and unacceptable", saying she supports imposing additional USA and United Nations sanctions.
Those people, Clinton said, could be open to persuasion. In a deluge of television advertisements in battleground states, Clinton and her Democratic allies have cast Trump as divisive and lacking the even temperament required of a president.
The rhetorical scuffle comes as the candidates head into the final two months of the campaign, with Trump trying to make up ground on Clinton before the November 8 election.
At the fundraiser, Clinton bemoaned the people she described as "deplorables", saying "unfortunately there are people like that". "Those are people we have to understand and empathise with, as well". "Now some of those folks, they are irredeemable, but they are not America".
Trump added: "How can she be president of our country when she has such contempt and disdain for so many great Americans?"
Shifting to the other half of Trump supporters, Clinton said many of those people feel like the government doesn't care about them and who just want change in any form.
Clinton made her remarks about Trump's supporters at a lavish event that featured performances from Barbra Streisand and Rufus Wainwright.
But, she added, it is "deplorable that Trump has built his campaign largely on prejudice and paranoia and given a national platform to hateful views and voices, including by retweeting fringe bigots with a few dozen followers and spreading their message to 11 million people".
Less than 24 hours after she made the comments at a private New York City fundraiser, Clinton said in a statement, "last night I was "grossly generalistic" and that's never a good idea".
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Trump's charge is that Clinton, a former USA secretary of state and former US senator, has profited from a "rigged" political system.
During a spirted speech focused on the LGBT community, Clinton also made a joke that referenced conversion therapy, type of counseling created to urge gay, lesbian, bisexual or transgender children to change their sexual orientation.
Speaking to donors on Friday night, Hillary Clinton outlined what she sees as the two types of Trump supporters.
There is little doubt that Trump has drawn support from extremist elements, including the so-called "Alt-Right" movement. He and retweets their offensive hateful mean-spirited rhetoric. But Democratic strategist Jim Manley, who supports Clinton, pointed out that Romney was talking about all voters, and Clinton was specifically describing Trump supporters.
"Wow, Hillary Clinton was SO to my supporters, millions of wonderful, hard-working people".
Friday's meeting and press conference, held at the storied New-York Historical Society, appeared staged to draw a stark contrast with Trump, whose comments this week raised further questions about his readiness to be commander in chief. "I think it will cost her at the Polls!"
Clinton spokesman Nick Merrill noted a previous speech in which she accused Trump of embracing a brand of US political conservatism associated with white nationalism and nativism known as the "alt-right" movement.
Some critics likened Clinton's observation to 2012 Republican candidate Mitt Romney's "47 per cent" comment in which he said 47 per cent of voters are dependent upon the government and would vote for President Barack Obama no matter what. "It's not surprising then they get bitter, they cling to guns or religion or antipathy toward people who aren't like them or anti-immigrant sentiment or anti-trade sentiment as a way to explain their frustrations", Obama said, in remarks later provided to the Huffington Post.
In that vein, his campaign has taken steps to distance the Republican nominee from his years-long propagation of the "birther" conspiracy theory that Obama was not born in the United States.
