WASHINGTON — Marilyn Cortez, a retired cafeteria worker in Houston with no health insurance, spent much of July in the hospital with Covid-19. When she finally returned home, she received a $ 36,000 bill that compounded the stress of her illness.

Then someone from the hospital, Houston Methodist, called and told her not to worry — President Trump had paid it.

But then another bill arrived, for twice as much.

Ms. Cortez’s care is supposed to be covered under a program Mr. Trump announced this spring as the coronavirus pandemic was taking hold — a time when millions of people were losing their health insurance and the administration was doubling down on trying to dismantle the Affordable Care Act, the law that had expanded coverage to more than 20 million people.

“This should alleviate any concern uninsured Americans may have about seeking the coronavirus treatment,” Mr. Trump said in April about the program, which is supposed to cover testing and treatment for uninsured people with Covid-19, using money from the federal coronavirus relief package passed by Congress.

The program has drawn little attention since, but a review by The New York Times of payments made through it, as well as interviews with hospital executives, patients and health policy researchers who have examined the payments, suggest the quickly concocted plan has not lived up to its promise. It has caused confusion at participating hospitals, which in some cases have mistakenly billed patients like Ms. Cortez who should be covered by it. Few patients seem to know the program exists, so they don’t question the charges. And some hospitals and other medical providers have chosen not to participate in the program, which bars them from seeking any payment from patients whose bills they submit to it.

Large numbers of patients have also been disqualified because Covid-19 has to be the primary diagnosis for a case to be covered (unless the patient is pregnant). Since hospitalized Covid patients often have other serious medical conditions, many have other primary diagnoses. At Jackson Health in Miami, for example, only 60 percent of uninsured Covid-19 patients had decisively met the requirements to have their charges covered under the program as of late July, a spokeswoman said.

Critics say the stopgap program is among the strongest evidence that Mr. Trump and his party have no vision for improving health coverage, and instead promote piecemeal solutions, even in a national health crisis. Mr. Trump had promised a plan to replace the Affordable Care Act by the beginning of August, but none has been announced and he and other Republicans barely mentioned health policy in their national convention last week.

For now, as tens of thousands of new coronavirus cases are reported each day in the United States — and as Democrats eagerly frame the election as a referendum on Mr. Trump’s handling of the pandemic and his efforts to wipe out the health law in the Supreme Court — the Covid-19 Uninsured Program is his best offer.

“This is not the way you deal with uninsured people during a public health emergency,” said Sara Rosenbaum, a professor of health law and policy at George Washington University.

The program has clearly paid what, in many cases, would be staggering and unaffordable bills for thousands of Covid-19 patients. In addition to hospital care, it covers outpatient visits, ambulance rides, medical equipment, skilled nursing home care and even future Covid vaccines for the uninsured, “subject to available funding.” It does not cover prescriptions once patients leave the hospital, or treatment of underlying chronic conditions that make many more vulnerable to the virus.

Health care providers in all 50 states had been reimbursed a total of $ 851 million from the fund as of last week — $ 267 million for testing and $ 584 million for treatment— with hospitals in Texas and New Jersey receiving the most.

But the Kaiser Family Foundation, a nonpartisan research organization, has estimated that hospital costs alone for uninsured coronavirus patients could reach between $ 13.9 billion and $ 41.8 billion, far more than what the program has paid out so far.

“The claims have just been so much smaller than anyone would have expected,” said Molly Smith, vice president for coverage and state issues forum at the American Hospital Association. “One thing we’ve heard a fair amount of is just serious backlogs and delays. But probably a lot of claims aren’t getting into the system at all because our members have determined they don’t qualify.”

The hospital association says that some hospitals have reported not submitting a substantial number of claims for their uninsured, with estimates ranging from 40 to 70 percent, because Covid-19 was not ruled their primary diagnosis.

“Either hospitals code inconsistent with ICD-10 rules,” said Tom Nickels, an executive vice president of the hospital association, referring to the diagnostic codes that hospitals use for billing, “or they don’t get paid even though the patient is clearly getting treated for Covid.”

Harris Health, a two-hospital public system in Houston, did not bill the federal fund for 80 percent of the roughly 1,300 uninsured Covid-19 patients it had treated through mid-July because many of them also had other medical problems — most often, sepsis, an overwhelming reaction to infection that causes blood-pressure loss and organ failure. In other cases, “an underlying health condition was the primary reason for hospitalization, but was exacerbated by the Covid-19 disease,” Bryan McLeod, a spokesman, said.

Nationally, the total average charge for uninsured Covid- patients requiring a hospital stay is $ 73,300, according to FAIR Health, a health care cost database, although they may be able to negotiate a lower amount.

​Reimbursements have varied widely with few obvious explanations; New Jersey providers, for example, have received $ 72 million in Covid treatment claims while those in neighboring New York have received half as much. Providers in hard-hit Texas and Florida, states that have not expanded Medicaid to cover more poor adults, have received $ 144 million and $ 53 million for treatment, respectively.

“It’s just not clear to me what’s going on,” said Jennifer Tolbert, director of state health reform at the Kaiser Family Foundation, who has looked closely at the program and its claims database.

Despite its limitations, some hospital executives said they liked the program because it paid Medicare rates, which are considerably higher than those for Medicaid, the government health insurance program for the poor, or any normal funding they would receive for charity care.

“This was a really progressive policy we were really surprised by, frankly,” said Dr. Shereef Elnahal, the chief executive of University Hospital in Newark, N.J., which has received $ 8.2 million for treating 787 uninsured patients with Covid, about a third of its coronavirus patients.

Unlike previous administrations during public health emergencies, Mr. Trump’s has not encouraged even temporary expansions of Medicaid — except for limited Covid testing — in states where the program covers few poor adults. It also declined to broadly reopen enrollment for Affordable Care Act plans once the pandemic began, although people who lose job-based coverage can enroll.

“You’re seeing a clash between enhancing Medicaid to allow it to cover the uninsured, versus providing a fixed amount of bailout money for providers who can figure out how to get to it,” Professor Rosenbaum said.

The Trump administration has not said how much money it will ultimately allot for the Covid-19 program. It is a small part of the Provider Relief Fund, which totaled $ 175 billion to start but has been spent down to $ 60 billion. Most of that money has helped health care providers cover financial losses during the pandemic.

A spokeswoman for the Department of Health and Human Services said in an email that it had not set a limit on how much of the fund would be spent on covering uninsured costs. She said the agency had no plans to change the program to cover patients for whom Covid is a secondary diagnosis.

For now, the confusion continues. Luis Fernandez, an oil industry worker in Houston who was laid off in January, had been uninsured for years when he got sick with Covid last month. He spent 16 days at Memorial Hermann Southwest Hospital and received bills totaling $ 85,000.

“I called the financial aid lady, told her I was on unemployment and she said, ‘What are you going to do?’” Mr. Fernandez, 33, said. “She wanted me to go back to work, like, tomorrow, so I can start paying them.”

A spokeswoman for Memorial Hermann said he had received the bill due to “a process error.”

“We do expect it to be paid since the primary diagnosis is Covid-19,” said the spokeswoman, Alex Loessin.

[Mr. Fernandez responded to a request for medical bills related to testing and treatment of the coronavirus. Read more about the project here.]

Ms. Cortez’s latest bill was for $ 76,363.

“This one scares me,” said Billy Cortez, her adult son, who shared a copy of the bill with The Times.

Stefanie Asin, a spokeswoman for Houston Methodist, said in response to questions from The Times that the program would cover all of Ms. Cortez’s costs but that the hospital system submitted bills to the program only once a month, creating delays. She described the process as lengthy, with “multiple steps and handoffs,” she said.

“We are looking to improve and accelerate this process for the benefit of our patients,” Ms. Asin said, “so they will not inadvertently get an unnecessary bill.”

Ms. Cortez still needs an oxygen tank, blood thinners and other medications that cost more than $ 500 a month. She also needs to start seeing a cardiologist because of all the blood clots she developed during her illness. Oxygen and outpatient visits related to Covid are supposed to be covered by the federal program, but nobody has told her that.

If Ms. Cortez lived in a different state, all her expenses would likely have been covered by Medicaid. But, like several million poor Americans, she is in the so-called coverage gap: ineligible for Medicaid because Texas is among a handful of states that have resisted expanding the program under the Affordable Care Act. And because of a glitch with the law, she and others are also ineligible for the subsidies that the law provides to help pay for private insurance.

For now, her son Billy is paying for her medications and praying she gets past the fatigue and coughing that still dog her. He said he was extremely grateful that his mother’s hospital bills would be paid, but worried about what the next two years, before she turns 65 and qualifies for Medicare, might bring.

“I feel like this isn’t going to be over any time soon,” he said.