In recent years, many but not all politicians have approached the issue of Brexit like you or I might approach an electric fence. Avoid getting too close and, if needs must, climb over it gingerly.
Burnham has no such reservations: he has repeatedly said in the last few weeks, let's talk about Brexit. He reckons the current deal between the UK and European Union isn't good enough.
Over the weekend he went further, suggesting he wanted an "honest discussion" about freedom of movement. Countries in the EU's single market have to sign up to freedom of movement, which lets people live and work in all the countries that are part of it. Given how big a deal immigration was for many in the EU referendum in 2016, even reopening the question of freedom of movement feels like a big deal.
On Monday morning, the prime minister appeared to add more detail, suggesting freedom of movement would only return in the UK having been sanctioned in a referendum.
Later, though, his team wouldn't repeat that, instead emphasising that Burnham wants to start a broad based conversation – without at this point committing to any possible outcome.
Some of his MPs felt this was a self-inflicted tangle he could have done without. Others see merit in Burnham's penchant for narrating his thoughts and possible direction publicly, thereby using his position to seek to dominate the political debate.
While the Labour Party's centre of gravity tilts towards a collective instinct that a closer relationship with the EU would be preferable to the status quo, there are differences of emphasis and indeed outright disagreements about how much closer and when would be wise.
Last week, Energy Secretary Miatta Fahnbulleh approved drilling in the Jackdaw field in the North Sea, about 150 miles east of Aberdeen.
This is principally a gas field and Fahnbulleh grounded her justification in providing the UK with greater energy security, at a time of war in Ukraine and in the Middle East.
The gas, she said, could be tapped into and serve British homes and businesses pretty much straight away, with the tax revenue put into an "energy wealth fund", with the aim being to invest in renewable and low carbon energy projects.
But what about the Rosebank field, 80 miles north west of the Shetland islands, with its focus on oil?
Labour figures in and outside government acknowledge this is a much, much harder sell to plenty in their own party and beyond. Some, who want to see the project go ahead, fear that the energy secretary is not likely to be convinced that it should.
It is in that context that we should see allies of the chancellor pointedly making it known that he is in favour of it.
His public language was quite telling on the day Jackdaw was approved, when he was talking about the Energy Wealth Fund: "It means the extra tax revenues that come from North Sea oil and gas can be directed into developing more clean energy projects for the future."
Note his reference to oil. Fahnbulleh said that Rosebank was "a different proposition" to Jackdaw.
Can those who worry, in particular, about the emissions from a new oil field, be won over by an argument grounded in this being a geopolitically pragmatic and environmentally credible part of transition to cleaner fuels?
Some Labour MPs hope so, but they know it's a tall order to persuade some, including some in government – which could, if it was approved, raise the question of ministerial resignations. Let's see.