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When New York City planners unveiled maps charting a 73-block area in the Bronx to be developed with housing, they called it “Cromwell-Jerome.”

But no one in that area seemed to know where that was.

The city seemed to make it up, said Alvaro Franco, who lives in the Bronx and quickly organized protests after hearing about the development plans. Mr. Franco said it sounded like “the same revisionism real estate agents in the city use to describe neighborhoods that already exist.”

City officials soon dropped the name. But the faux pas demonstrated the jitters among residents and the pressures on Mayor Bill de Blasio as he tackles the piece of his income-inequality agenda that, unlike his fights for preschool education and higher wages, promises to change many of the city’s neighborhoods.

The plan, which calls for 80,000 new apartments, mostly for households with annual income of less than $69,000, requires an extraordinary amount of diplomacy, even with the mayor’s allies. Neighborhood groups and their City Council representatives, who must sign off on any rezoning, are anxious about taller buildings, more people and gentrification. Labor unions want assurances that they will have a bigger role in construction, even though it drives up costs.

Perhaps the most vocal complaints are coming from affordable-housing groups that want the new homes that are designated as affordable to go to the poorest residents, as opposed to a mix of income levels, and that want to ensure that current residents will not be displaced as people with higher incomes move into the neighborhood and make it more upscale.

While the activists acknowledge that they have enjoyed more access to a post-Bloomberg City Hall, they have also been conducting frequent protests there. At a demonstration in March, protesters yelled “Slow it down!” as Mr. de Blasio passed by them.

“It hasn’t been easy for either side,” said Maritza Silva-Farrell, a spokeswoman for Real Affordability for All, a coalition of more than 50 housing and tenant groups. “It’s hard when you hear that a community is going to be rezoned and you don’t know what that means.”

Administration officials have played down the anxiety, but there is obvious concern, and there are signs that the resistance is wearing on them. In an interview, Mr. de Blasio said he planned to go out into communities to “talk about the plan directly to the people.”

“We have to show people in neighborhoods all over the city that this is a very different approach to development and one that protects their interests,” the mayor said.

Alicia Glen, the deputy mayor for housing and economic development, is known among affordable-housing advocates and developers as a brusque-talking, no-nonsense negotiator. But she began tearing up in a recent interview as she described a housing plan she said was aimed at helping lower-income New Yorkers stay in the city.

“It actually makes me cry saying it,” Ms. Glen said. “No other city has ever tried to do this. And the fact that people don’t get that just makes me really crazy.”

Administration officials are seeking to remake poor corners of the city into economically diverse neighborhoods, and their blueprint would reserve 20 percent for the poorest households (annual income of less than $43,150 for a family of four); 58 percent for families making up to $69,000; and 22 percent for those with income of up to $142,400.

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Andrew Berman, executive director of the Greenwich Village Society for Historic Preservation, organized a protest on the City Hall steps just before the meeting. “The argument about affordable housing is totally specious,” he said in an interview.

Mr. Berman said the administration risked changing the character of neighborhoods for a few affordable housing units.

“Look, if we were talking about changing the rules for bigger buildings with 100 percent affordable housing, this would be a different conversation,” he said. “We’re being asked to give up a lot and get little, if anything, in return.”

Some of the new units will be in buildings where all or most of the units are affordable, but others will be in buildings where most of the units are market rate. Those projects help the mayor’s goal and require relatively few city subsidies. But neighborhood groups fear the encroachment of taller buildings and, in some cases, more affluent residents, who can cause quick gentrification and push out the poor and existing businesses, altogether transforming neighborhoods.

Williamsburg, a Brooklyn neighborhood that seemingly changed overnight, is almost a dirty word among both administration officials and housing groups because many residents were left feeling the area did not get as many affordable units as it could have from new construction.

Any displacement of current residents could also have racial overtones, since many of the neighborhoods in the mayor’s plan are mostly minority. The now-renamed Jerome Avenue corridor in the Bronx (Cromwell, the name of a neighborhood street, was dropped) is 65 percent Latino and 30 percent African-American, and the median household income is $26,226, compared with $51,865 citywide.

The city has tools to help preserve neighborhoods as affordable, including financial incentives for landlords to keep rents low and a new legal fund to help tenants fight landlord harassment and evictions. But the most important anti-displacement measure, housing activists insist, is to have the new affordable units, and their rents, reflect the existing income levels of rezoned areas. That could be more costly for the city, however, since cheaper rents require more subsidy.