Rishi Sunak emerged as Conservative leader from the wreckage of the Liz Truss premiership exactly a year ago, vowing to “fix” things. But 12 months later, some of the problems facing him and his party are as intractable as ever.

The prime minister, dubbed by supporters as “Rishi the problem solver”, has so far failed to answer the biggest conundrum of all: how can the Conservatives win a fifth consecutive election?

Sunak’s party trails Labour by an average of 17-18 points, while his own personal approval ratings have been falling since the spring and are at record lows. Two by-election humiliations last week confirmed the scale of the problem.

One former cabinet minister said: “We are absolutely devoid of vision and hope. There’s no way of turning it around — we are heading for a huge defeat.”

Sunak became Tory leader on October 24 2022 and entered Downing Street a day later. The third prime minister in two months, he promised “economic stability and confidence” after the chaos of Truss and Boris Johnson.

Markets were quickly reassured and the technocratic Sunak in early 2023 set himself five tests: cutting inflation, cutting debt, delivering growth, cutting hospital waiting lists and tackling migration in small boats.

He resolved the bitter post-Brexit row with the EU over Northern Ireland, restored relations with Emmanuel Macron, the French president, and signed a defence pact with the US and Australia.

But the polls refused to budge and his advisers decided a new approach was needed. “Fixing things wasn’t enough,” said one Sunak strategist. “The next election is going to be about ‘change’ and Rishi can be that change.”

Convincing the country that he represents “change” after 13 years of Conservative government is a tall order and Sunak’s first efforts appear to have been largely ineffective.

Paul Goodman, editor of the ConservativeHome party activists’ website, said Sunak’s Tory conference speech this month — in which he scrapped the northern leg of the HS2 rail line and promised reforms to A-level examinations and a phasing out of smoking — had left voters cold.

“He seems to have decided to do things that interest him,” Goodman said, noting that Sunak had always “hated HS2”. But he added: “There’s no sign that the voters are expressing engagement with these issues or that his speech made any positive difference.”

The Tory by-election defeats this month in the formerly safe seats of Mid Bedfordshire and Tamworth have fuelled a sense of fatalism among many Conservatives, who feel the party is sliding out of power.

“In January the view in the party was that there was a route through,” said one Tory supporter of Sunak. “We had a sensible, pragmatic, properly Conservative prime minister and maybe we could be back in contention. The mood now is depressed.”

Sunak’s chief of staff, Liam Booth-Smith, has told any colleagues who think that the next election is unwinnable to quit their jobs now and in the prime minister’s team there is still a belief that things can be turned around.

The King’s Speech on November 7, setting out Sunak’s final legislative package of the parliament, is the next big event, followed by chancellor Jeremy Hunt’s Autumn Statement on November 22.

The fiscal backdrop is grim and Hunt has said that tax cuts are “virtually impossible” now. His task will be to take the “difficult decisions”, which could include spending and real-terms benefit cuts, to try to create some space for tax cuts in a pre-election Budget next year.

Among the potential tax cuts brought forward are a reduction in stamp duty, inheritance tax or a raising of the threshold for the 40 per cent income tax band. But they will not disguise the fact that since Johnson’s 2019 election victory, taxes have risen by an average £3,500 per household.

Many on the Tory right are hoping Hunt will be among those moved in a cabinet reshuffle, although Downing Street has made it clear that this will not happen imminently. “The prime minister and chancellor are working very closely on the Autumn Statement,” Sunak’s spokesperson said.

If Hunt was replaced by a “fresher face” before next year’s Budget, some Tory MPs believe Sunak could promote the neophyte energy security minister Claire Coutinho, a former Merrill Lynch and KPMG executive. “She’s Rishi’s kind of person,” said one Tory MP.

Other names touted by Sunak’s allies for promotion in a reshuffle include relative Westminster newcomers Laura Trott, pensions minister, and Richard Holden, transport minister.

Few Tory MPs give credence to the idea that rebels might seriously try to oust the prime minister. Sunak, therefore, has some political space and Sir Robert Buckland, a former justice secretary, urged him to use it.

The prime minister should embrace “more political risk” and demonstrate a “sense of purpose and future vision”, particularly regarding housebuilding, aspiration and the tax burden, said Buckland.

“He might be halfway through his premiership. He hasn’t got time on his side.”