“I hear about, ‘Well we have the funds to do it,’” she said. “So why haven’t you done it? Take an empty lot that is just sitting there, that no one is using, put a building on it.”

“No one wants to take these people off the streets because of the way they look,” she said, sitting with her daughter, Kaylee, 18, who was born while Ms. Hayward was homeless. “Well at one point, I looked like that. At one time, I was that homeless person.”

Mayor Eric M. Garcetti said the passage last year of the housing measure, known as HHH, demonstrated the breadth of support for confronting the homelessness problem. He said the battle in Boyle Heights was just the latest in a long history of development disputes that have roiled the city.

Image Christina Hayward with her children, Kaylee and Ben, at their apartment complex in Compton. Ms. Hayward has lived there with her husband and children for eight years. The family was once homeless. Credit... Emily Berl for The New York Times

“L.A. said loud and clear we need to build more housing, we need to build more affordable housing and we need to build more housing for the homeless,” he said. “But that shouldn’t be interpreted as a blank check for any project. As you are seeing with this project, people always have strong opinions.”

“The challenge to anyone who says ‘no’ is they have to find a place to say ‘yes,’” Mr. Garcetti said.

The 49-unit project, called Lorena Plaza, would rise on a dusty lot across the street from a Pizza Hut and a cemetery. About half of the units would be for residents with mental illness. The site had been a staging area for construction of the Metro Gold Line, which now rumbles by. Lorena Plaza would be next to El Mercado, a Mexican-American community institution — with shops and a top-floor restaurant with mariachi bands — whose owners have been at the forefront of the opposition.