Some officials are backing off requirements that people wear masks inside businesses, as cities, counties and states — left to devise their own guidelines — run into limits on their ability to maintain public health precautions with stay-at-home orders easing during the coronavirus pandemic.

The issue pushed a small Oklahoma city into the national spotlight this weekend, after leaders quickly withdrew a mandate to don masks inside reopened stores and restaurants, citing threats of violence and physical abuse directed at employees. The mayor of Stillwater apologized to businesses for putting them in a dangerous position, as some people responded virulently to the new rules.

“We don’t have the kind of police force that can go out and try to deal with every single one of the people who may not be willing to wear the masks,” Mayor Will Joyce said Sunday on MSNBC. “And so it’s been a struggle [to] make people understand that wearing that face covering is an easy and an effective way to help slow the spread of this virus.”

Joyce’s comments came the same day that Ohio Gov. Mike DeWine (R), an early proponent of strict statewide social distancing, said he had reversed course on requiring Ohioans to wear masks because people “were not going to accept the government telling them what to do.”

“It just wasn’t going to work,” he said on ABC News’s “This Week.” “You got to know what you can do and what you can’t do.”

Messaging from federal officials on masks has been changing and at times contradictory. Health authorities began recommending last month that all Americans cover their faces in public after previously calling it unnecessary. White House coronavirus task force coordinator Deborah Birx on Sunday called protesters who defy stay-at-home orders and crowd together without masks “devastatingly worrisome.”

But while some cities and counties mandate masks and even threaten $1,000 fines amid a patchwork of reopening strategies, leaders including President Trump and Vice President Pence have conspicuously not worn them in certain public appearances. (Pence said Sunday that he should have covered his face during a much-criticized visit to the Mayo Clinic recently.) And mask requirements are stirring the same kind of politically charged resistance to broad restrictions on Americans’ lives that has popped up in angry rallies at state capitols and governors’ residences, sometimes encouraged by the president.

“There are a lot of mixed messages out there,” Joyce said Sunday on MSNBC, adding that his city reopened along with the rest of Oklahoma to keep people “on the same page” despite reservations about whether the timing was right for Stillwater.

“I think it would be best from a nationwide perspective,” he added, “if we could have … a unified message.”

For some, defiance to mask-wearing has a conservative bent. To many Trump supporters, declining to wear a mask is a visible way to demonstrate “that ‘I’m a Republican’ or ‘I want businesses to start up again’ or ‘I support the president,’ ” Robert Kahn, a law professor at the University of St. Thomas in Minneapolis who has studied Americans’ attitudes toward masks, told The Washington Post last month.

“Masks will quickly become the new normal in blue states, but if social distancing continues through 2022, the mentality among Republicans could well change, too,” he predicted. “If I can go to work and the cost of marginal improvement in my life is wearing a mask, maybe Americans of both parties do accommodate ourselves to it.”

The mayor of Stillwater, an independent, said the city has encouraged mask-wearing in public for weeks. But havoc broke out the moment officials tried to make doing so mandatory.

Joyce amended his emergency declaration on the same day it took effect “in response to concerns voiced by business proprietors and citizens,” according to a news release. The anger directed toward store employees started in the first three hours businesses were open and included a threat of gun violence, the release said.

“I hate that our businesses and their employees had to deal with abuse today, and I apologize for putting them in that position,” Joyce said in a statement. “I am not the kind of person who backs down from bullies, but I also will not send someone else to fight the battle for me.”

In Stillwater, many people objecting to the mask requirement cited their belief that the rule is unconstitutional, City Manager Norman McNickle said in the news release first reported by the Stillwater News Press. McNickle said wearing a mask is a minor inconvenience that protects the person wearing it and anyone the person encounters.

“It is unfortunate and distressing that those who refuse and threaten violence are so self-absorbed as to not follow what is a simple show of respect and kindness to others,” McNickle wrote, saying that officials could not in good conscience put store and restaurant employees in danger.

Stillwater’s stores are still asked to at least encourage customers to wear masks, and specific businesses can choose to have more restrictive requirements.

Stillwater Police Chief Jeff Watts said in a statement that his department was not pulling over drivers for not wearing masks, responding to complaints about people not wearing face coverings in businesses, or ticketing residents who are not wearing masks in public. Watts said customers and employees of certain kinds of businesses, including salons, barbershops and tattoo parlors, must still wear masks.

On Sunday, Joyce — whose Twitter account features a picture of him in a cloth face covering — posted photos of vitriolic messages he has received criticizing him both for the original mask requirement and for backing off of it. “You didn’t think we would react to your TYRANT behavior with threats?” one message reads.

“This is a hard time for everyone,” he wrote in a tweet thread, offering up his email address to anyone who wants to talk. “We may not agree on the way forward. But we can be thoughtful and compassionate to our neighbors. We can try to see and understand their perspective.”

Aaron Gregg contributed to this report.