Keir Starmer does not have a problem with women, the transport secretary, Louise Haigh, has said, but she admitted the government had made “missteps”.

Haigh, one of a number of female cabinet ministers who had been close to Starmer’s former chief of staff Sue Gray, defended the prime minister after Gray left her post to be replaced by his campaign director, Morgan McSweeney.

Starmer had also been accused of having a woman problem by the former Labour MP Rosie Duffield, who quit the party last month.

But Haigh said the prime minister had promoted a number of women. “I don’t think the prime minister has any problem working with women,” she 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.”