The White House denied on Sunday that Donald Trump “caved” to Democrats in reopening the federal government despite not receiving any funding for a border wall – a deal Trump had vowed he would never make.

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Hundreds of government facilities began to come back online following the 35-day partial closure, meanwhile, as hundreds of thousands of furloughed employees awaited back pay the administration promised would arrive by the end of the week.

“You can only be so happy because you just have to know that it could happen again,” Rachel Malcom, whose husband serves in the US Coast Guard in Rhode Island, told the Associated Press. “We’re going to be playing catch-up, so I don’t want to overspend.”

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Federal facilities were reopening on a staggered timeline. In Washington the popular Smithsonian museums on the National Mall and the National Zoo were to remain closed until Tuesday. Facilities such as Fort McHenry national monument in Baltimore, Maryland, the defense of which in the War of 1812 inspired the American national anthem, opened at the weekend.

A rash of delays at airports from understaffed security checkpoints and reported holes in the air traffic control system on the east coast had subsided, after sharpening the pressure on Trump to make a deal on Friday.

Internal Revenue Service employees, border security workers, corrections officers and others required to work during the shutdown without pay were told on Sunday they would be compensated in short order, as would employees barred from their jobs during the shutdown.

“Some of them could be [paid] early this week, some of them may be later this week,” the acting White House chief of staff, Mick Mulvaney, told CBS’s Face the Nation. “But we hope that by the end of this week all of the back pay will be made up, and of course the next payroll will go out on time.”

But for some workers living paycheck-to-paycheck, the road back to normalcy is blocked by obstacles such as charges for missed debt payments, damaged credit ratings and doubts about future job security.

Those doubts were stoked by Trump’s threat on Friday to shut the government down again “if we don’t get a fair deal from Congress” before temporary funding runs out on 15 February.

Mulvaney said Trump was serious about the threat, despite an estimate by the economist Stephen S Fuller of George Mason University that the shutdown cost the Washington DC regional economy $1.6bn in January and that 112,500 federal contractors – as opposed to full employees – would not receive back pay. The estimate was first reported in the Washington Post.

“Yeah I think he actually is,” Mulvaney told CBS, asked whether Trump was prepared to repeat the shutdown. “Keep in mind he’s willing to do whatever it takes to secure the border. He does take this very seriously.”

While the US public overwhelmingly blamed Trump for the shutdown, the Republican party sought to blame Democrats, a sign of potential trouble ahead as negotiations between the sides resume.

“Democrats have held our government hostage for weeks but thanks to President @realDonaldTrump’s leadership, the government will reopen and federal workers will be paid in the next few days,” the national GOP tweeted.

The message was echoed by Mulvaney, who denied that Trump had lost the shutdown showdown to Democrats.

“No, I think what you’ve seen here is the president seeing an opportunity,” he told Fox News Sunday. Democrats indicated to Trump that they would agree to money for border security in the next round of budget talks now under way, Mulvaney said.

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The Democratic West Virginia senator Joe Manchin, a key potential crossover vote for any Republican budget proposal, said Democrats were looking for a “holistic approach” to immigration reform – which could indicate room for compromise or, given the vexed history of such attempts, daunting complications ahead.

“We have proven it’s hard for us because with the president and White House and the legislature, we have locked horns on this thing – no wall, all wall, halfway in between,” Manchin told CBS. “Let the professionals tell us what it takes to keep us safe.”

A minority of the US public supports Trump’s plan to erect a wall on the southern border with Mexico, and Trump’s average disapproval rating spiked during the shutdown from 52% to 56%.

Critics of the border wall plan say it would be a waste of money to address a problem the president grossly exaggerates and which in any case would be better addressed by spending on a combination of technology and personnel.

Trump campaigned for the White House on a promise to build the wall and have Mexico pay for it. He has claimed a new trade deal will produce sufficient savings to meet the second half of his promise. Most analysts doubt it.