But the fatal attack on Sunday at the Acacia-run shelter — where city officials say SERA provides security — has led to renewed calls for New York to more urgently address concerns about the nonprofit. Acacia operates 750 individual family housing units as well as four shelters that house 550 homeless adults in the city, according to the department.

“There is a real problem here and we need to do something before another life is lost,” said Councilman Ben Kallos, chairman of the City Council’s contracting committee.

Mr. Kallos, a Democrat, says he plans to call on Monday for a Council hearing regarding Acacia’s practices.

Since the nonprofit came under investigation, Mr. Kallos said several current and former residents in Acacia shelters had shared stories of their troubling experiences: Some told him they were forced to share rooms with other residents who had threatened to kill them or had previously injured them. Others said Acacia staff members had threatened to evict them if they called the police about the conditions inside the shelter.

“One of the hardest problems is that the people in these shelters and making these reports are those who the system and society might not treat as credible,” Mr. Kallos said. “But in light of what happened yesterday, that seems less and less the case.”

A spokesman for Acacia said on Sunday night that staff members at the shelter immediately notified the police after the attack and were working with police investigators. “We are deeply saddened by the tragedy that took place,” he said. “Guards patrol this 103-unit shelter around the clock and violence is extremely rare, as this is the first violent incident recorded there all year.”

For decades, the city’s shelter system has been notorious for the dangerous conditions at some of its sites, particularly at intake and assessment centers — the system’s front door. Those threats are often the reason homeless people prefer to stay on the streets or in subway cars, instead of at shelters.