Kamala Harris and Donald Trump spent Sunday trying to shore up political support among what they perceived to be must-have voting blocs with polls showing them locked in a tight 5 November presidential race.

With election day less than a month away, the Democratic vice-president attended a Black church in Greenville, North Carolina, as part of her campaign’s “souls to the polls” push. She later exalted the way communities – especially in the western part of the state – were coming together after damage from Hurricane Helene in late September, especially the way “people who have the least give the most”.

Her Republican opponent, meanwhile, was in Arizona – looking for Black and Latino support as he seeks a second presidency, after a rally in California a day earlier.

Both candidates are attempting to get a decisive edge among votes who have not yet decided who to support. Surveys show that early voting, which tends to favor Democrats, is down 45% from previous election years – a sign that there may be millions of undecided voters.

Trump has now switched from condemning early voting as a Democrat plot to engineer his defeat to Joe Biden in 2020 to urging people to vote early and by mail.

A recent ABC News-Ipsos poll showed that support was split down gender lines, with women voting 60-40 to Harris and men breaking for Trump by a similar margin.

Trump needs white women, who supported him in a greater numbers in 2020 than in 2016 – but also Black men. On Sunday, he argued that his fellow former president Barack Obama’s call last week for Black men to support Harris based “solely on her skin color, rather than her policies” as “deeply insulting”.

Democratic Georgia senator Ralph Warnock on Sunday told CNN, “Black men are not going to vote for Donald Trump in any significant numbers.” But his fellow Black Democrat Jim Clyburn, a South Carolina congressman, told CNN, “Yes, I am concerned,” about Black men voting for Trump. Separately, former president Bill Clinton was urging voters in rural Georgia to get behind the Democratic ticket.

A New York Times poll published Sunday found that Harris is underperforming the last three Democratic candidates for the White House among Latino voters.

The election may come down to fractional increases in support for each. An NBC News poll released Sunday showed the candidates in a “dead heat” nationally at 48% support. The poll found that voters are reassessing Trump’s first term more favorably – but also that voters view reproductive rights as a top motivating issue, which could hurt the former president after three of his US supreme court appointees eliminated the federal right to abortion.

A CBS News poll, also released Sunday, found that the presidential race is more than just two conflicting ideologies – but about a fundamental disconnection.

For instance, most Trump supporters said relief for victims of Hurricanes Helene and Milton wasn’t reaching affected people – while Harris supporters indicated it was. Trump supporters said the economy was bad; Harris supporters said it was good. Trump’s voters said US-Mexico border crossings were increasing; Harris’s voters said they were down.

Trump’s voters, especially the men, said gender equality efforts had gone too far; Harris voters said not far enough. But both agreed that social media was untrustworthy and had made it harder to find things to agree on and to tell fact from fiction.

Each poll contained positive signs for Harris, including a five-point advantage on “looking out for middle class” (ABC); abortion being “#1 motivating issue” (NBC), with Democrat up 19 points on the issue over Trump (New York Times); Trump’s Latino support at the same level from 2020 (CBS), and also Harris matching Biden in 2020 with Black voters.

But the response to the two hurricanes that the south-eastern US recently continued to dominate Democrats’ campaign. On Sunday, Biden was scheduled to survey damage inflicted on Florida’s Gulf coast by Milton, where he would announce $600m in funding for damaged electrical grids.

Response to hurricanes remains Democrats’ political preoccupation. Harris’s rally Sunday came amid the intense politicization of the speed of federal disaster response to Helene.

In North Carolina, Harris appeared to be looking to defuse hurricane politics while also calling out false information that spread after Helene.

Crises, she said, “have a way of revealing the heroes among us, the angels among us, and of showing us all the best of who we are ... heroes who do not ask the injured or stranded whether they are a Republican or a Democrat, but who simply ask: ‘Are you OK?’”

And yet, Harris said: “There are some who are not acting in the spirit of community, and I am speaking of these who have been literally not telling the truth, lying about people who are working hard to help the folks in need, spreading disinformation when the truth and facts are required.”

That came as the Wall Street Journal reported that some of the earlier response to Helene had come in the form of Patriot Front, an organization that the Anti-Defamation League has concluded is a white-supremacist group – and that was using misinformation as a recruiting tool.

With Arizona, Nevada and Georgia potentially leaning for Trump, and Pennsylvania, Michigan and Wisconsin potentially leaning for Harris, the loss of North Carolina would cost Trump 16 electoral college votes needed to reach the winning threshold of 270. The state narrowly voted for Trump in 2020.

The Republican House speaker, Mike Johnson, told CBS’s Face the Nation on Sunday that he would deny Harris and Biden’s call to bring Congress back to Washington to approve more disaster relief funding after the hurricane.

“It can wait,” Johnson said, pointing to $20bn in additional disaster funding that had recently been approved. He claimed only 2% of that funding had been distributed. As soon as states have assessed and calculate their “actual needs”, and submitted them, “Congress will meet and in bipartisan fashion, we will address those needs.”

Johnson accused the Federal Emergency Management Agency (Fema) of being “slow to respond”. He said: “They did not do the job that we all expect and hope that they will do, and there’s going to be a lot of assessment about that as well in the days ahead.”

But with Harris’s support appearing to slip in recent weeks, including after a series of TV appearances, there are reports of growing tensions between her campaign and Biden’s White House. The president cancelled a trip to Germany to concentrate on the hurricane response. But he is now reported to have rescheduled the trip for Friday.

According to Axios, Biden aides remain wounded by the president being pushed out of his re-election bid amid questions about his age. He is 81 – only three years older than Trump.

Harris’s team believed Biden upstaged her by holding an impromptu press briefing while she held a rally in Michigan.

Biden on Sunday was expected to meet with Florida governor Ron DeSantis, with whom Harris was feuding earlier in the week. An aide to Harris, 59, told the outlet that the president’s team are “too much in their feelings”.