On a conference call this spring, New York City officials introduced the company that was stepping in to help handle a major humanitarian crisis, as homeless migrants continued to arrive in the city by the thousands.

Several people on the call, which included advocates for the migrants, had a concern: “This company doesn’t have any kind of immigrant experience,” said Dawedo Sanon, a volunteer coordinator at the Columbia County Sanctuary Movement. “It makes no sense.”

About three months after that call, the company — DocGo, a medical services provider — is facing a litany of complaints, including accusations that its representatives or subcontractors deceived and threatened migrants and, most recently, that it may have interfered with a police investigation into two allegations of sexual assault at migrant shelters it runs in motels upstate.

DocGo’s troubles come at a particularly precarious moment in the migrant crisis, as the administrations of Mayor Eric Adams and Gov. Kathy Hochul face mounting criticism from the public — and from each other — over how they are handling the response.

New York City has put DocGo in charge of key tasks, including helping to manage its main intake center at a Manhattan hotel. But concerns about the company’s performance are creating additional headaches as the city struggles to respond to an influx of people that seems unlikely to let up soon.

The police in the Buffalo suburb of Cheektowaga, where some migrants relocated from New York City are staying, say they are investigating the interactions DocGo had with the victims and suspects in two alleged sexual assaults before the police arrived. In one instance, a migrant was accused of assaulting another migrant staying at one of the hotel shelters DocGo operates, and in another instance a migrant was accused of assaulting a DocGo subcontractor.

The town’s police chief, Brian Gould, said investigators were trying to determine whether the company’s actions constituted witness tampering. “We’re concerned that when we try to conduct an investigation, if somebody has already spoken to the suspect and the victim, that is absolutely going to affect our ability to prepare the cases,” he said.

DocGo’s chief executive, Anthony Capone, said the company was “devastated by what happened” and was fully cooperating with law enforcement. He also told local government officials and advocates for migrants in Albany last week that he was working to rectify problems that had surfaced since New York City gave the company a no-bid $432 million contract, which is financing migrant relocations from crowded city shelters to communities upstate, stretching from Rockland County to Buffalo.

But the police investigation has increased pressure on the publicly traded company at a time when Ms. Hochul, the city comptroller, Brad Lander, and other officials in and around Albany were already calling for increased scrutiny and oversight of DocGo. After the assault allegations, New York City temporarily halted migrant relocations to Buffalo, and Ms. Hochul has sent National Guard troops to the hotels where they are staying.

“I’m not satisfied as I’m standing here, but I’m going to be reviewing this,” Ms. Hochul told reporters in Albany when asked recently about DocGo’s performance. “Instances have arisen, been reported on, that are deeply concerning to me.”

On Friday, her administration’s Office of Temporary and Disability Assistance, which is reviewing DocGo’s conduct, released a statement saying the city had failed to provide the company’s contract as required and that the agency was “prepared to take additional actions to compel this disclosure if necessary.”

DocGo saw its fortunes skyrocket with Covid testing and vaccination contracts for New York City during the pandemic, but the company began doing migrant care only late last year, when the city enlisted its help to handle the flow of asylum seekers from the southern border to Manhattan’s Port Authority Bus Terminal.

The pivot came just in time. As DocGo was warning investors its Covid testing money had all but dried up, the company landed its lucrative $432 million contract with the Department of Housing Preservation and Development to house, transport and provide care to migrants. The contract took effect in early May, records show.

A couple of weeks later, after perplexed advocates for migrants wondered on the conference call why a medical services company would be put in charge of migrant relocations, New York City officials explained that DocGo, valued at a billion dollars and traded on the Nasdaq, would be able to quickly scale up and provide the range of services migrants required.

“The city is notorious for not paying on time,” the city’s chief engagement officer, Betsy MacLean, said on the call. “DocGo is a bigger company. They have the cash flow to be able to sustain those kinds of gaps, and still pay subcontractors.”

Yet as she spoke, the city’s contracting agency was in the midst of a lobbying campaign, which would drag on for weeks, to give DocGo a cash advance to pay for start-up costs for its work caring for migrants.

New documents obtained by The New York Times show that the agency asked the comptroller, Mr. Lander, to approve a payment of $4 million to the company, though the city rarely extends such cash advances, the comptroller’s office said.

In an application for the advance to Mr. Lander’s office on May 17, an assistant commissioner for the Department of Housing Preservation and Development, Merrick Reid, wrote that “the contractor will not enter into an agreement with the City without advance payment(s) for services.” Mr. Reid also said the city had “exhausted efforts to find a contractor” that would do the work without cash up front.

Mr. Lander’s aides pushed back, asking why, as one official in his office put it in an email to the agency, “a large, publicly traded, for-profit vendor warrants advance payment for its services.”

In an interview, Mr. Lander said that only three times since the influx of migrants prompted a city emergency declaration had agencies requested advance payments for migrant services. One of them was a nonprofit with cash flow issues.

His office denied the request on July 18.

“Based on what we now know, it seems clear that my staff made the right decision,” Mr. Lander said. “Reporting that’s come to light raises serious questions about this vendor specifically and also exposes the greater vulnerability of emergency procurement.”

William Fowler, a spokesman for the Department of Housing Preservation and Development, said DocGo had helped increase capacity for migrants by providing shelter and services to 4,000 people in 28 hotels both in New York City and upstate. The agency said DocGo was using its own resources to open new hotels as it awaits payment.

Nine days after the denial of the cash advance, the city’s public hospital system announced at its board meeting that DocGo had been awarded another migrant management contract without competitive bidding — a deal worth up to $311 million — by amending a pre-existing contract the company had with the hospital system, records show.

A spokesman for the hospitals, Adam Shrier, said emergency procurement procedures were required to “secure the immediate support of dedicated, proven contractors” to help the city navigate the crisis.

The maximum value of the two migrant-related deals — over $740 million combined — dwarfs the company’s total reported revenue in 2022 of $440 million. Though it did not receive the cash advance, the company ultimately agreed to go forward with the work.

The Times reported last month that many of the migrants DocGo bused to Albany — as part of Mayor Eric Adams’ “decompression strategy” to relieve pressure on the city — said they were treated like prisoners in a halfway house and provided with deceptive work and residency papers while in the contractor’s care.

DocGo said it had discontinued use of the documents, some of them on what appeared to be fake city letterhead. The company also said it had made changes to its security team after The Times reported that one security guard threatened to harm migrants and another was target shooting with an air rifle in the parking lot of one of the hotels where migrants are staying.

But scrutiny of the company has continued. Earlier this month in Albany, city, county and state officials criticized DocGo, saying they continued to hear complaints ranging from subpar food to unmet medical needs.

Assemblyman John T. McDonald III, a Democrat whose district includes parts of Albany County, said local nonprofits had taken on some of the work of helping migrants, without being paid.

“The question is: What in God’s name is DocGo doing?” Mr. McDonald said.

Mr. Capone, the company’s chief executive, last week met for about an hour and a half with local officials and groups helping migrants in Albany, according to two people who attended the meeting but were not authorized to speak about it. He promised to give more information about people arriving by bus, to upgrade the food service for migrants, to provide them with unlimited city bus passes and to retrain caseworkers, the two people said.

The previous week, on Aug. 11, authorities in Erie County arrested a migrant from Central Africa on charges of sexually assaulting one of DocGo’s contracted caseworkers. It was the second such arrest in a week’s time.

After the arrests, Mr. Gould, the police chief, said officials were looking into whether DocGo and partner entities had tried to interfere with their investigations.

“Did they encourage somebody not to report something? I don’t know,” the chief said. “That’s what we’re trying to get to the bottom of.”

Mr. Capone said DocGo was “working closely with local police, has fully cooperated throughout the investigation, and will continue to do so.”

“Our heart goes out to the individual directly impacted,” Mr. Capone said, adding that the safety and well-being of the company’s employees and those in its care “are paramount.”

After the second assault was reported to authorities, Erie County Executive Mark Poloncarz called Mr. Adams and asked him to immediately halt any further relocations to Buffalo. He said the mayor agreed to a pause.

Besides the police investigation, Mr. Poloncarz, a Democrat, faulted the company for failing to conduct adequate health screenings and said its caseworkers appeared to have little or no experience dealing with migrant populations.

“What we thought was going to happen is not happening under DocGo,” Mr. Poloncarz said. “I’m disappointed, to put it mildly.”