His promises represented an effort to regain momentum at a time of steep challenges for Britain, but some pressing problems, like the National Health System, defy easy solutions.LONDON — With Britain’s health system and economy both in acute distress, Prime Minister Rishi Sunak on Wednesday laid out a series of promises to restore the country to prosperity and well-being, putting his own political future on the line by challenging Britons to hold him to account.Mr. Sunak’s pledges, delivered in a sweeping speech that echoed a State of the Union address by an American president, represented his effort to grab back momentum after a period in which he replaced a discredited predecessor, Liz Truss, and mopped up after her calamitous foray into trickle-down economics.“No tricks, no ambiguity,” Mr. Sunak said to a polite audience in East London. “We’re either delivering for you or we’re not.”Among the promises, the prime minister said he would cut inflation in half, reignite the economy and reduce waiting times in emergency rooms — ambitious goals for a government that has so far been largely a hostage to a series of disruptive events.But some of Britain’s most pressing problems, like its overwhelmed and investment-starved National Health System, defy easy solutions. Even with more funding, Mr. Sunak said, “people are waiting too long for the care they need,” citing the ambulances lining up in front of hospitals that are short of beds for patients.Budget strains and a cost-of-living crisis have triggered widespread labor unrest, with nurses walking off hospital wards and railway workers shutting down trains. The government is expected to announce new anti-strike legislation, but Mr. Sunak conceded the difficulty of making deals with multiple unions, even though polls show Britons generally support the workers.“I don’t think anybody thinks a 19 percent pay rise is affordable,” he said of the nurses’ wage demands.A crowded King’s Cross station in London last week. Industrial actions by railway workers disrupted travel over the holidays.Hollie Adams/Getty ImagesBeyond that, the British economy is also likely to deteriorate further before it bottoms out and begins to recover. Mr. Sunak acknowledged that sobering reality, noting that many Britons were looking ahead to 2023 with “apprehension.”For Mr. Sunak, who has come under criticism for his below-the-radar style, the speech was an effort to offer much-needed reassurance and to present an image of a sturdy leader at the helm. With two years to go before he must call an election, he billed his five promises — which also included cutting public debt and stopping the perilous flow of migrant boats across the English Channel — as yardsticks with which to judge his government.Understand the Political Situation in BritainA Political Test: Rishi Sunak, who took over as prime minister with the hope of restoring stability to a government in turmoil, is facing formidable political and economic challenges.Worker Strikes: Crippling strikes across multiple industries have Britain’s Conservative government facing a “winter of discontent,” just as a Labour government did 44 years ago.Migrant Crossings: Under growing pressure to curb the arrival of migrants in small boats on the English coast, Mr. Sunak announced plans to tackle Britain’s backlog in asylum claims and to fast-track the return of most Albanians seeking refugee status.Selling Austerity: With an elite pedigree and a privileged lifestyle, Mr. Sunak must now persuade ordinary Britons that they should support his government through a painful ordeal of tax increases and spending cuts.Eschewing the ideological extremism of Ms. Truss or the have-your-cake-and-eat-it-too optimism of her predecessor, Boris Johnson, Mr. Sunak struck a nuts-and-bolts tone. Characteristically, his most widely promoted initiative was a plan for all school children to study mathematics until the age of 18.“One of the biggest changes in mind-set we need in education today is to reimagine our approach to numeracy,” said Mr. Sunak, a line that would have been unlikely to turn up in a speech by Mr. Johnson.Still, some experts said there was less to some of Mr. Sunak’s promises than met the eye. The Bank of England has already projected that the inflation rate, currently 10.1 percent, will decline to roughly half that by the end of 2023. That downward trend, in any event, has less to do with fiscal than monetary policy.Mr. Sunak’s pledge to “grow the economy” by the end of the year was noteworthy, given that it is now likely shrinking. But he offered few prescriptions for how the government planned to do that. Britain has struggled with lackluster productivity and stagnant growth for more than a decade.Shopping for groceries in London last November, when inflation hit a record 12.4 percent.Andy Rain/EPA, via Shutterstock“Growth will return, almost certainly in the next year or so, but that’s a very low bar,” said Jonathan Portes, a professor of economics and public policy at Kings College London. “I would point out that Truss set an explicit growth target of 2.5 percent, so Sunak is being much less ambitious.”Mr. Sunak, a 42-year-old onetime investment banker who served as chancellor of the Exchequer under Mr. Johnson, faces a huge task improving public services. The N.H.S., one of Britain’s most revered institutions, suffered during years of austerity under Conservative-led governments, and was then battered by the pandemic.Jill Rutter, a senior fellow at the Institute for Government, a London-based research institute, said that by the time of the next general election, Mr. Sunak will need to be able to show the British public that things were improving and that it would therefore be a risk to eject him from power.“Most public services were looking pretty fragile at the time of the pandemic, and the pandemic then piled problems on top of them, including big treatment backlogs in health and exhaustion among the work force” Ms. Rutter said. Those problems, she said, were “compounded by inflation and a big squeeze on public sector pay.’’Most of these underlying weaknesses will remain, even if the government resolves the pay dispute with nurses and ambulance drivers. “Even if Rishi Sunak and Jeremy Hunt write a big check to the National Health Service, that doesn’t solve the capacity problem quickly,” she said, referring to the current chancellor.Similarly, Mr. Sunak has a limited number of options for reviving the economy even if inflation tapers off and interest rates stop rising. Last fall, Mr. Hunt reversed the tax cuts announced by Ms. Truss, replacing them with a raft of tax increases and spending cuts. The reversal restored Britain’s tarnished reputation in financial markets, but at a cost to economic activity at home.Nurses striking outside St. Thomas’ hospital in London last month. Kin Cheung/Associated PressMr. Sunak also needs to manage divisions within his fractious Conservative Party, while knowing that Mr. Johnson harbors ambitions to return to Downing Street, if given the opportunity.“One of the problems for Sunak is that his party is so all over the place that, on a whole range of issues, if he goes one way, he’ll alienate a bunch of them and if he goes another, he’ll alienate another bunch,” Ms. Rutter said.Any attempt to solve labor shortages by relaxing immigration rules, for example, would prompt opposition from a right-wing faction within the Conservative Party, as could any compromise with the European Union over post-Brexit trade rules for Northern Ireland.One of Mr. Sunak’s most immediate challenges is cutting down the flotilla of small boats carrying asylum seekers across the channel. On Wednesday, he pledged new laws that would stop the crossings, but provided neither a timetable nor evidence of how deporting illegal migrants would stop the influx.By sketching out his priorities for the next year, however, Mr. Sunak will hope to quiet critics who claimed that he has stayed out of the spotlight as alarm spread over the state of the health service, and as the latest wave of strikes paralyzed parts of the country.The leader of the opposition Labour Party, Keir Starmer, was scheduled to make a speech on his agenda on Thursday. Mr. Sunak’s hastily scheduled appearance prevented his rival from exploiting a political vacuum to build on Labour’s polling lead over the Conservatives, now more than 20 percentage points.Like Mr. Sunak, Mr. Starmer is regarded as an uninspiring public speaker. His critics accuse him of excessive caution and of failing to articulate how he would change the country as prime minister.For Mr. Sunak, the challenge is more immediate but no less daunting: convince skeptics that he measures up to the job of prime minister at a time of converging crises. More