The transport secretary, Louise Haigh, stated that although the prime minister had elevated several women, Keir Starmer did not have an issue with women. However, she acknowledged that the government had committed “missteps”.

Following Sue Gray's resignation and the appointment of Morgan McSweeney as Starmer's campaign director, Haigh, one of several female cabinet members who had been close to the previous chief of staff, refuted the notion that the prime minister had a problem with women.

In addition, Starmer was accused of having a woman issue by Rosie Duffield, a former Labour MP who left the party last month.

“I don’t think the prime minister has any problem working with women,” Haigh told Sky News. “If you look at the women he has around him, the first female chancellor [Rachel Reeves], Angela Rayner – the cabinet is gender balanced. We have more female Labour MPs than there are Tory MPs in total.

“So I don’t think any sense that the Labour party has a problem with women – or the prime minister – is evidenced by the facts of us.”

Gray, who left her role after weeks of headlines about infighting in No 10, was an “exceptional public servant”, Haigh said. Gray will be Starmer’s envoy on nations and regions and is widely expected to get a peerage.

Haigh told Times Radio it was “right that the operations of Downing Street are reviewed and that they properly support the delivery of government”.

But in a tacit acknowledgment that there had been some difficult months for Labour, she said it was expected that a new government would take time to work.

“This is a young government, there is bound to be missteps in the first few months. Very few of us have served in government before. We have got 14 years of opposition and 14 years of a juggernaut to turn around,” she said.

Asked if there would be more mistakes, Haigh said: “No government is perfect, and I am not going to sit here today and promise you there is going to be no mistakes made.”

Starmer’s new chief of staff is expected to make radical changes to the operation, and No 10 sources said “nothing was off the table”.

That could mean another shake-up of teams inside No 10, resolving the pay dispute with special advisers and adding to the political side of the operation with more appointments.

“The day-to-day machinery of government doesn’t work properly. Morgan has a different view on how to do things. He’s nothing if not the agent of change,” said one senior source.

“He will have to be radical. Nothing is off the table. We’re expected to be running a powerful G7 economy yet sometimes it’s all quite small and analogue. There has to be a better way of doing things.”

The prime minister’s official spokesperson said Starmer was “completely focused” on delivery.

“He’s made the right changes to ensure that we have strengthened the No 10 operation to deliver the change the country voted for.”

But the spokesperson appeared to accept that Gray’s departure meant Starmer recognised the internal problems in Downing Street.

“I think it’s right to reflect on the first weeks and months in office to ensure that you do have the right structures in place going forward to deliver change for the country.”