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 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.”