The San Francisco Board of Supervisors on Tuesday approved one of Mayor Ed Lee’s final requests, to declare a citywide emergency and fast-track construction of shelters for homeless people.

But several supervisors contested and ultimately delayed voting on a second proposal from the mayor, who died early Tuesday, to add showers and laundry facilities at an office building on 440 Turk St. where the city’s Department of Homelessness and Supportive Housing is headquartered.

Supervisor Jane Kim, whose district includes the 440 Turk St. building and the Tenderloin neighborhood that surrounds it, said the plan was a misuse of the term “emergency.”

She noted also that neighborhood residents had objected to the plan, which would bring additional services for the homeless to an area that already has plenty, some neighbors say.

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Those criticisms led to a fierce debate among supervisors. Some sided with Kim and her Tenderloin constituents, saying the building was poorly conceived and that the city had unfairly forced it on residents.

Others feared that homeless services were being demonized and that Lee’s legacy of helping people in need had died along with him.

“I hope his passing doesn’t end this work,” said Supervisor Hillary Ronen, a progressive who ran against Lee’s policies and then collaborated with him to open new Navigation Centers — residential homeless shelters with services — in her district.

The board put off voting on the 440 Turk St. plan until January but upheld a companion measure to streamline the city’s construction bidding process so that additional homeless shelters can be built quickly. Lee had said the city needed to open several new facilities this winter to meet his goal of getting 1,000 more people off the streets.

Also Tuesday, the board passed a measure co-sponsored by Lee and Supervisor Ahsha Safai to create an assistance fund for tenants displaced from buildings that are deemed uninhabitable. It was partly inspired by the Ghost Ship warehouse that burned last year in Oakland, and by the basement laundry in the Excelsior where two dozen people lived until January.

Additionally, Supervisors Aaron Peskin and Katy Tang introduced a June ballot measure to limit how much the city spends on budget set-asides — pots of money earmarked for certain programs — when the projected budget deficit exceeds $200 million. San Francisco currently has 19 set-asides, more than any other jurisdiction in the country. Los Angeles, by contrast, has two, and San Diego has one.

Each of the city’s set-asides began as well-intentioned, voter-approved measures to isolate funding for items like street trees, parks, children’s services and public libraries. Over the years, they piled up, Tang and Peskin said, eating up $1.2 billion of the city’s budget in the current fiscal year.

The mandatory spending requirements have created imbalances in the city’s budget, bloating some programs and agencies while others struggle with limited funds, Peskin said.

Safaí and Peskin introduced a June ballot measure to create a new Department of Livable Streets, which would handle parking, traffic, pedestrian rules and other neighborhood issues that were formerly overseen by the San Francisco Municipal Transportation Agency.

The new department would send decisions to the supervisors for final approval. SFMTA would still handle Muni.

Rachel Swan is a San Francisco Chronicle staff writer. Email: rswan@sfchronicle.com Twitter: @rachelswan