A coalition of some of the mayor’s liberal allies is set to start a campaign Thursday to create more homes for the city’s poorest residents, posing an additional challenge for the administration as it develops its own housing plan.

The group, dubbed Real Affordability for All, plans to rally on the steps of City Hall and present a report it says shows a yawning gap in units aimed at residents making less than $41,000 a year for a family of four.

Real Affordability for All, which consists of about 50 tenant groups, anti-poverty advocates and unions, is a type of political group emerging as a powerful force under Mayor Bill de Blasio: activists who can secure meetings with senior administration officials while also being willing to take to the streets. Others in the category are the Working Families Party and labor unions

In this case, the group’s demands could make it more difficult for the mayor to meet his target of building or preserving 200,000 units of low-cost housing, with his plan due by May 1.

The reason: Units targeted to low-income residents typically require more subsidies from the city, because the developer must make up for the lower rents.

“One of the things that we’ve been pushing the administration on is that this isn’t a drive toward a number, this is a drive toward affordability,” said Jonathan Westin, executive director of New York Communities for Change, a coalition member.

The report says policies pursued by Mayor Michael Bloomberg created more than enough housing for those making around $41,000 a year for a family of four, but failed to help about 700,000 residents in households making less.

Ismene Speliotis, executive director of the Mutual Housing Association of New York, said her research, using 2011 census data, found that there were more than 250,000 units affordable to some 119,000 households making 51% to 60% of the area median income, which amounts to about $40,000 to $57,000 a year for a family of four.

Rental housing for families is defined as affordable when they can pay for it with 30% of their income.

In contrast, about 168,500 units were available to roughly 620,000 households making less than 30% of the area median income, or about $25,000 a year for a family of four.

Ms. Speliotis traces the discrepancy to policy that limits the maximum rents for housing projects that receive federal tax credits to 60% of the area’s median income (in New York, roughly $57,000 a year for a family of four).

Unless the city provides subsidies to make it possible, she said developers will tend to build to that maximum income level.

City officials provided census data that suggest while there is a significant need for very low-income units, the study incorrectly tabulated thousands of affordable apartments, making the need look higher than it is.

The data also suggest that low-income households—making roughly $33,000 to $65,000 a year—have a significantly higher rate of rent burden than very-low income households making less than $41,000 who have more access to government subsidies and benefits to help close the financial gap.

Ms. Speliotis and Mr. Westin have met with city officials, including Deputy Mayor for Housing and Economic Development Alicia Glen, Director of Intergovernmental Affairs Emma Wolfe and Vicki Been, the new head of the Housing Preservation and Development Department, according to people familiar with the matter.

“It’s going to be our mission—and our challenge—to broaden the income levels at each project, with a particular focus on low-income New YorkersWe want vibrant, mixed-income communities that truly reflect the city’s diversity,” said a spokeswoman for the mayor.

Affordable-housing experts and developers said they saw demand across the spectrum and the city must decide how to best allocate its resources.

Martin Dunn, president of Dunn Development Corp., an affordable-housing developer, said he received “huge numbers of applications” from people around 60% of the median income level, but said he saw an even higher need below and above it.

He said he supported targeting more units to lower income levels, but said it “takes more resources.”

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