When Tania Fernandes Anderson left the City Council’s chamber last week for another engagement on her calendar, she and her colleagues had okayed sending a multibillion dollar operating budget, covering a vast array of city services, to Mayor’s Wu’s desk.

By the time she returned, the budget, initially passed on a 8-4 vote, was on its way back to the committee she chairs, Ways and Means, as two councillors backtracked amid resistance to potential cuts to city departments.

That budget, which totaled $4.2 billion, included about $75 million in amendments from councillors, who pitched for more money for tuition assistance for Boston Public Schools students, a salary increase for municipal police officers, and an expansion of the city’s tree canopy, among other items on a long list. But the funding had to come from somewhere, and in this case, it was a $42 million cut to the Boston Police Department, according to At-Large Councillor Michael Flaherty, who argued that his fellow councillors were taking a “meat cleaver” to public safety.

Flaherty and Councillors Ed Flynn, Frank Baker, and Erin Murphy were all part of the “no” votes before the reversal of two councillors: Brian Worrell of Dorchester and Gabriela Coletta of East Boston.

Coletta voiced concerns about layoffs as a result of the cuts and changed her vote to “no.” Just before the Council meeting ended, Worrell, who is the vice chair of the Ways and Means Committee, changed his vote to “present,” denying the budget the majority needed to move ahead. He did not explain his switched vote.

Councillors are in their second go-around with enhanced budgetary powers approved by voters in 2021. Before the charter amendment was okayed, councillors could only approve, reject, or reduce a budget proposed by the mayor. Now they can amend the mayor’s proposal, as long as their total does not exceed the total proposed by the mayor. That leaves them with the ability to amend up to $1.8 billion, or 42 percent of the fiscal 2024 budget that Wu has put forward.

The amendment also created new deadlines for councillors to act: They had to vote on or before the second Wednesday in June, which is June 14 this year. A majority – seven votes – is required to send it off to the mayor, who then has seven days to approve, veto, or make her own amendments. Then it heads back to the Council, which needs a two-thirds majority (9 votes) to override a veto or mayoral amendments, which has to be done before July 1, the start of the 2024 fiscal year.

If they don’t act by June 14, according to Fernandes Anderson, the mayor’s budget proposal goes into effect. “It is do or die,” she said.

During last year’s budget process, councillors cut $65,814 from the budget and never reappropriated it. In another move, they sought to cut police overtime, but they were blocked by Wu, who proposed a smaller police budget cut to the tune of $1.2 million. The independent Boston Municipal Research Bureau, in an analysis afterwards, said the council overall “redistributed within the operating budget only $9.9M, or 0.6%, of appropriations the City Council had authority to amend.”

Chairing the Council’s budget process for the second time this year, Fernandes Anderson said last week that she did not reject any councillor’s amendments, leaving it up to the mayor to say no. “She’s the mother. She has to take care of the city,” Fernandes Anderson said.

At working sessions on Monday and Tuesday, councillors appeared to be making some progress in whittling down the size of their amendments in an effort to reach a compromise ahead of the June 14 Council meeting, as well as reaching some truces.

Councillor Flaherty, while opposing cuts to the police department, said he was willing to give up his $1.5 million amendment funding a vehicle for the Boston Fire Department.

But Councillor At-Large Julia Mejia told her colleagues that she supported keeping the amendment in. “You heard that, Flaherty?” she called out across the room. “I got your back there, kiddo.”