Burnham told reporters his own speech to the UN General Assembly later would reflect on the decade since the Brexit vote and the “turbulence comes when outside actors seek to manipulate, to distort facts, influence the British democratic process”.
“Going into the next decade, we need to be clear-eyed about that and become stronger and more on the front foot about challenging that,” he said.
Asked if he would rule out making a pledge to rejoin the EU at the next general election, Burnham said: “Our focus at the moment is on a UK-EU summit which we hope will take place later this year.
“My immediate priority is to build on the good work that Keir [Starmer] did in rebuilding bridges, because those bridges were broken in the early part of this decade.”
He added: “What we have got to do is now go to the next level in terms of reaching practical agreement that will further boost the British economy, and those issues are the substance of the summit alongside EU requests around youth mobility.”
He said questions about rejoining the EU were for “another day”.
Speaking at the Liberal Democrat conference in Brighton on Monday, the EU’s new ambassador to the UK was asked whether the UK could rejoin the EU or be granted associate membership.
Jukka Salovaara said: “The doors are open but it’s very much up to the UK public to identify and decide on its level of ambition when it comes to deepening cooperation with the EU.”
Last September, Burnham said he would like to see the UK back in the EU in his lifetime.
Earlier this year, he said Brexit had been “damaging” but “the last thing we should do right now is re-run those arguments”.
The EU has recently suggested Canada could become the bloc’s first “associate member”.
Asked if this could be an option for the UK, Burnham said: “The European Union made it clear that that wasn’t open to us.”
Additional reporting by Brian Wheeler