Boris Johnson has written to ministers and heads of Government departments reminding them that there is “no place for bullying” after an investigation into the Home Secretary found she had shouted and sworn at staff.

There was further Downing Street fall-out on Friday when Sir Alex Allan, the Prime Minister’s adviser on ministerial standards, walked out after Mr Johnson overruled his conclusion that Priti Patel breached the ministerial code and opted to stand by his Home Secretary.

Only the week before, Mr Johnson had seen two of his most senior advisers, de facto chief of staff Dominic Cummings and director of communications Lee Cain, announce their departure from Number 10.

Downing Street at the weekend did not deny suggestions that Mr Johnson had tried and failed to convince Sir Alex to tone down his conclusion – published during anti-bullying week – that Ms Patel’s behaviour amounted to bullying as he found instances of shouting and swearing.

Issuing a written ministerial statement, the Prime Minister acknowledged that Sir Alex had concluded that Ms Patel’s behaviour could “on occasion” be described “as bullying in terms of the impact felt by individuals”, but added that he had “full confidence” in the Cabinet minister and that he considered “this matter now closed”.

Mr Johnson has, in a joint letter with Cabinet Secretary Simon Case to civil service permanent secretaries and those in the ministerial ranks, called for “strong working relationships” to be “at the heart of this Government”.

In a letter published on the Government website, the pair said: “The relationship between ministers and civil servants is one of the foundation stones in our system of government.

“Given the unprecedented challenges we currently face as a nation, relationships of mutual trust and respect between politicians and their officials are paramount. This includes keeping internal conversations private.

“We should all feel able to speak freely and honestly about matters of state. We should also feel able to speak constructively about things that are not working, so that we can fix them together promptly.

“There is a particular duty on ministers and permanent secretaries to create jointly across Government a culture which is professional, respectful, focused and ambitious for change and in which there is no place for bullying.

“Never has it been more important that we work as one team. Let’s put strong working relationships between ministers and their officials at the heart of this Government.”

Following the publication of the report, the Home Secretary apologised and said there were “no excuses” for what happened but highlighted Sir Alex’s assessment of her awareness.

She told broadcasters last week that “any upset that I’ve caused is completely unintentional and at the time, of course it says it’s in the report, that issues were not pointed out to me”.

Labour has called for the report into Ms Patel’s behaviour to be published in full.