“I get calls all day, every day,” he said. “I’ve basically joined the help desk.”

Philadelphia is a case study in the simple-but-not-easy task of helping tenants with the rent. Social programs are often a partnership in which cities provide funding and lay out rules but delegate the execution to quasi-governmental nonprofit organizations like Mr. Heller’s. Like most places, Philadelphia isn’t close to satisfying the need for help. But through rounds of rejiggering and three phases of funding — each with its own maze of rules and requirements — Mr. Heller’s group built a team to distribute aid, whittled down the processes that delayed it and ultimately concluded that the best way to help was the most straightforward: Give the money directly to renters.

“There’s a societal belief that poor people can’t spend money the right way, and I think it’s important to start questioning that assumption,” Mr. Heller said.

Almost from the moment the pandemic spread across the United States, advocacy groups have warned that the economic fallout could cause mass displacement of low-income tenants. In response, more than 400 state and local governments have used money from the federal CARES Act to set up funds to cover at least $4.3 billion in rental assistance — money that has helped tenants pay their bills and landlords stay current on their mortgages, according to a database set up by the National Low Income Housing Coalition, a policy group.

But now many jurisdictions are reporting trouble spending it, and with barely two weeks left in the year, they are on pace to have more than $300 million left over, according to the coalition’s database. In a pattern that predated the pandemic, the programs have been complicated by bureaucratic hurdles, competing budget demands and a reluctance among landlords to take part.

There was shifting federal guidance on how CARES Act money could be spent. States passed legislation that piled local rules on top of the federal rules. Each layer was ostensibly created to improve programs — preventing fraud, making sure the money went to the neediest tenants — but added numerous hurdles for both tenants and landlords, and in the end cost time.