America's mayors know their constituents hold them accountable for homelessness, but many don't feel they have the tools or power to fix things, a brand new survey says.

Why it matters: While homelessness has become more acute during the pandemic, city leaders say they lack the money, staff or political support needed to make a meaningful difference.

Driving the news: A poll released Tuesday by the Menino Survey of Mayors found that 73% of mayors believe that voters hold them highly accountable for addressing homelessness, but only 19% think they have much control over the issue.

• Pessimism runs deepest among mayors in the Northeast, with just 7% saying they have lots of control.
• Limited funding is by far the top barrier to reducing homelessness, mayors say, followed by public opposition to new housing and shelters.
• Among other factors cited are insufficient human and social services, lack of coordination among responsible agencies, and evictions.

The big picture: It's hard to gauge whether homelessness has gotten better or worse during the pandemic. HUD's most recent national point-in-time census count was in January 2020 — before the pandemic started.

• Last year, many big cities called off their census counts, normally conducted in January.
• This year, while HUD isn't allowing cities to forgo the annual exercise, some are postponing their counts until February.
• At the same time, cities from Boston to Minneapolis are clearing out big homeless encampments, moves that typically stir backlash on all sides.

The number of homeless families may actually have gone down, a result of expanded pandemic-era tax credits and unemployment benefits, Nan Roman, president & CEO of the Alliance to End Homelessness, tells Axios.

Details: 126 mayors were surveyed between June and August by the Boston University Initiative on Cities' Menino Survey of Mayors.

• Almost a third said they had no staff dedicated to homelessness.
• Nearly a quarter said they relied on the police to deal with the issue.
• "Mayors of high and low housing cost cities and across all regions of the country hold remarkably similar views," per the report.

The bottom line: There are billions of dollars in the CARES Act and the American Rescue Plan Act to combat homelessness and help people and families who are at risk, but they will need to be deployed wisely.

• "There really is the potential to make a big dent in homelessness with that money, and the question is, are we going to?" Roman says. "That's a good question for mayors."
• "Mayors often don't have the money in their hands, but they influence everybody who does have the money."

Go deeper: Big cities call off 2021 homeless counts due to pandemic safety