There’s an argument that any leader would struggle with the conditions Mr. Sunak inherited: high inflation, increased borrowing costs and low growth. Across the world, incumbent governments of all stripes are finding their time is up — whether it’s the center-left Labour Party in New Zealand or the right-wing populist Law and Justice party in Poland. When Mr. Sunak has found success, it’s been by making his own weather. His renegotiation of the Northern Ireland protocol, an especially vexed post-Brexit arrangement, showed maturity and won him a brief popularity bounce.

Yet economic difficulties have been stubborn. Mr. Sunak, a former chancellor, was picked by lawmakers because of his economic credentials — and he has managed to win back some market confidence. But the government is still boxed in. The right of the party, including the outspoken Ms. Truss, wants tax cuts. Mr. Sunak won’t budge until inflation is down, which is not happening quickly enough. Facing a winter of high bills, Britons will be feeling the pinch for some time to come.

But Mr. Sunak’s biggest challenge is the length of time his party has been in power. The Conservatives, plagued by scandal, have overseen a country where discontent is legion: A survey taken this summer found that three-quarters of people in Britain believe it is becoming a worse place to live. After 13 years of Tory rule — the same amount of time New Labour, under Tony Blair and Gordon Brown, enjoyed in office — the other side can’t be blamed for Britain’s woes.

Mr. Sunak’s attempts to overcome this fundamental problem are twofold. First, he has accepted that the country is not working and needs to change. His five priorities — halving inflation, stopping the boats carrying migrants across the Channel, cutting National Health Service waiting lists, growing the economy and reducing debt — are designed to reflect key voter concerns. But many are pessimistic that all the goals can be achieved. Continuing health worker strikes, for example, signal that unhappiness with the state of the N.H.S. is unlikely to subside ahead of the election.

His second move is more ambitious. In a bid to shake off the baggage of previous Tory governments, Mr. Sunak is trying to depict himself as the change candidate. He has axed David Cameron’s pet project, a high-speed rail network linking the Midlands and the North, and scaled down the net-zero commitments embraced by Boris Johnson and Theresa May. The goal is to show him as a man of action with his own convictions, someone prepared, as he recently put it, to “be bold.” But running against your party’s own record is tricky, and it is already causing resentment among colleagues who served in previous administrations.