Greater Manchester Mayor Andy Burnham says he is concerned about the “balance” of the cabinet after a reshuffle sparked by Angela Rayner’s resignation for under paying stamp duty.

She left her roles as housing secretary and deputy prime minister, as well as deputy leader of the Labour Party, the latter of which has triggered a contest to find her replacement.

Burnham told Sunday with Laura Kuenssberg that process would be an opportunity to have a “discussion about the internal management” of the party and that somebody from the north of England would be “great”.

Defence Secretary John Healey said there was a “good new team in place” and called on them to “go up a gear”.

Rayner’s exit has sparked a major reshuffle with David Lammy replacing her as deputy prime minister, Yvette Cooper moving to foreign secretary and Shabana Mahmood following her as home secretary.

Among those to lose their job was Leader of the House Lucy Powell.

Asked if he was worried about whether the new-look cabinet included the right balance of people, Burnham said: “I am concerned about the balance, and I think you know, we need to use the deputy leadership contest to discuss some of these things.”

He said the leadership needed to to the party more, pointing to internal rows over winter fuel payments and disability benefits, adding that MPs opposed to the government’s plans should not be “punished”.

He continued: “I see good people, good MPs, losing the whip, people like Rachael Maskell. That doesn’t seem fair to me. It didn’t happen in the governments I was in, in Gordon Brown’s government or Tony Blair’s government.

“You know, we need a different style here, so that everyone is included and we pull together.”

His words come ahead of a deputy leadership contest which some have predicted could highlight divisions between Downing Street and the left of the party. The deputy leadership is separate to the deputy prime minister role, although Rayner held both simultaneously.

Asked who should take the job, Burnham suggested Louise Haigh or Lucy Powell, both of whom are MPs in northern England, in order to “counter that London centricity”.

Healey told the BBC the reshuffle was Sir Keir Starmer resetting his team and “saying to each and every one of us you’ve got to go up a gear, you’ve got to deliver”.

He added their job was “to demonstrate to people that we can as government change their lives, make them feel better off and make them feel more secure”.

Healey said Rayner had been an “inspiration to many” and particularly to working-class women, adding the government would “miss” her, adding: “Any government is stronger with Angela Rayner but we’ve got a good new team in place and that’s our job.”

But he defended Sir Keir’s “clear, swift action” and said she was “not victim”.

Labour MP Emily Thornberry said she was considering running for deputy leader but said it was “a question of what can I bring to it”.

“I mean I hear what is said about the deputy leader should come from Manchester, and obviously the mayor of Manchester would say that.

“I’m sure that people would say they should come from Wales or Scotland.

“What’s important is what you bring… do you have the strength, do you have the experience, can you actually make a difference?”

Labour’s governing body is due to meet on Monday to discuss details of the deputy leadership election, according to LabourList.