Sir Keir Starmer’s government has not yet been in place for 100 days. By all accounts, things are not going well.

The Labour leader and his top ministers have had to backtrack after taking freebies totalling many thousands of pounds for clothing and entertainment. Sue Gray, the ex-civil servant he recruited as chief of staff, has accepted a salary larger than the prime minister while standing accused of cutting the pay of more junior special advisers coming into government.

The biggest announcement the government has made yet – cutting winter fuel payments for most pensioners – was poorly presented, coinciding with big pay rises for public sector trade unionists and leading to a rebellion by Labour MPs. So far, eight of them have either been suspended or resigned the whip. Labour is dropping in opinion polls. Meanwhile, the prime minister often seems defensive and belligerent when interviewed and at a loose end at important gatherings.

It is still early days. None of these teething troubles directly threaten a government that commands an overwhelming majority in the House of Commons. But there is universal agreement ranging from Sir Keir’s friends to his political foes that he needs to get a grip on running the country, starting with appointing the best people as his senior advisers and officials.

The historian Sir Anthony Seldon, the author of a series of books on prime ministers in 10 Downing Street, warns “Starmer needs to act quickly. He has been naive and complacent on staff appointments. Get it right now and he can fly.”

This week a major opportunity presented itself. The cabinet secretary, Sir Simon Case, at last announced he will step down at the end of this year. The man or woman who fills his shoes will be vital in properly establishing and relaunching the way the UK is ruled by the new government.

According to the official advert: “The cabinet secretary and head of the civil service is the most senior civil servant in the UK and the principal official adviser to the prime minister and Cabinet.”

The salary is £200,000 a year, also more than the prime minister, plus a hefty 28.7% pension contribution. Applicants have until 11.55pm on 20 October to get in their CV, a 1000-word statement, a diversity questionnaire and a declaration of interests.

Sir Simon’s departure had been long expected. Some blame the prime minister for not forcing the vacancy sooner. Cabinet secretary is the most important of an unprecedented number of unfilled posts among Starmer’s top advisers, also including principal private secretary (PPS), his personal civil service aide, and national security adviser. Without them, few are surprised he has not got on top of being prime minister.

Sir Simon has been absent for long periods due to a serious neurological illness. In truth, he never really settled in as the respected boss of some half a million civil servants. Still only 45, he was much younger than most cabinet secretaries when installed in 2020 by the chaotic Boris Johnson government after Mark Sedwill was unceremoniously pushed aside. Sir Simon came from being private secretary to Prince William. He had also been PPS to Theresa May.

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Rightly or wrongly, senior Labour figures felt he struggled to serve the new government. An unproven suspicion lingers on the Labour side that he may have been behind leaks damaging to Sue Gray.

The successful working of the UK’s constitutional machinery depends on an impartial cabinet secretary. His main job (so far they have all been men) is to ensure the civil service delivers the government’s programme effectively.

The cabinet secretary must also advise the prime minister whether their plans make sense, are acceptable and conform to expected ethical standards. In the perceptive TV comedy series Yes Prime Minister the catchphrase of cabinet secretary Sir Humphrey Appleby was: “Is that wise, prime minister?”

By the time they get the top job, most cabinet secretaries have worked for governments and ministers across the political spectrum. Sir Gus O’Donnell was a close adviser to John Major, Tony Blair and Gordon Brown. In 2010 he oversaw the transition to what became the Conservative/Liberal Democrat coalition between David Cameron and Nick Clegg.

When Lord O’Donnell’s successor Sir Jeremy Heywood died prematurely of cancer in 2018 Blair, Brown, Cameron, Clegg and Theresa May led the official mourners. Sir Jeremy’s advice was hung on by successive prime ministers. If anything, he was too helpful and got too close. He and David Cameron were both keen to involve the disgraced former businessman Lex Greensill with government business.

Past history and present tensions suggest that it would be a mistake to appoint Olly Robbins as the cabinet secretary. He is widely seen as Sue Gray’s preferred candidate. Downing Street might benefit from some creative tension. Besides, Robbins’ experience, including as a Brexit negotiator, is better suited to national security adviser.

Sir Keir has inherited a mess created by Rishi Sunak. In the dying days of the last government, Mr Sunak attempted to promote his national security council adviser, Tim Barrow, to US ambassador and to replace him with General Gwyn Jenkins. The Labour opposition cried foul successfully. As a result, both jobs and the people involved with them are now up in the air. A decision on the new Washington ambassador is due after the American election in November.

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Prime ministers who appoint friends and cronies exclusively as their key advisers tend to come a cropper, as the five recent Conservative leaders found out eventually. Far better to bring in someone of proven administrative expertise as cabinet secretary.

As usual, there are some highly rated senior male civil servants, with experience heading government departments, who could fill the role, led by Jeremy Pocklington, permanent secretary, or “perm sec”, at DESNZ, the energy department, and Sir Peter Schofield, department for work and pensions’ perm sec.

The new government has made much of having the first female chancellor of the exchequer. If Sir Keir fancies appointing the first woman cabinet secretary there is a rich and colourful field to choose from.

Sarah Healey, perm sec at DHCLG (communities) and formerly the culture department, is widely respected, as is Tamara Finkelstein at DEFRA. In Lady Finkelstein’s case, some in Labour might balk at handing the top job to the wife of Danny Finkelstein, the prominent Conservative peer and Times newspaper columnist.

Dame Antonia Romeo, currently in charge of the Ministry of Justice, has held a number of senior civil service jobs. During Liz Truss’s short-lived tenure, she was briefly appointed Treasury perm sec. Her earlier, high profile, high fashion, approach to being UK consul general in New York brought her into conflict with the British embassy in Washington DC.

Former female civil servants who might want to return to government include Dame Melanie Dawes, currently cautiously low profile at the media regulator Ofcom, and Baroness Minouche Shafik, former perm sec at the department for international development and deputy governor of the Bank of England. She has just finished as president of Columbia University in New York City. Then there is Helen McNamara, who had a bruising time as deputy cabinet secretary during the COVID pandemic.

It is not yet known who of those above will put their names forward. One of the most widely tipped names is not applying. Sharon White, of John Lewis, Ofcom and the Treasury, could have been the first woman and the first black person to be cabinet secretary. She has decided to sit on the selection panel instead, alongside Gus O’Donnell.

The final choice of the next cabinet secretary will be made by the prime minister. The pressure is on Sir Keir to think out of his comfort zone if he is to be guided from the missteps of the first 100 days into four years of competent and ethical government.