It is just coincidence that a very public mishap by Zack Polanski arrived at the same time as new polls showed Labour in the lead. But it did neatly illustrate the point that while Andy Burnham is enjoying a political honeymoon, the Green leader’s now seems over.

This is not to say the Green party in England and Wales is too worried about the arrival of a new prime minister, or that there is necessarily much evidence yet that they should be.

One of two national polls this week showing Labour overtaking Reform for the first time in more than a year did also show the Greens’ score falling by three percentage points. But the second showed them constant at a healthy 13%, well above pre-Polanski levels.

Publicly and privately, people in the party say that while Burnham is obviously a better communicator that Keir Starmer, they see no sign of him being notably bolder on policy in ways that will tempt Green supporters to look at Labour.

“It’s still very early days, and we don’t know what will happen with Burnham,” one senior party figure said. “But on areas like the cost of living and drilling in the North Sea, it doesn’t feel like much will change.”

However, an internal debate about Polanski’s prolific and largely solo social media efforts has started after he shared an Instagram post that included someone wearing a T-shirt intimating that they would welcome Nigel Farage’s death.

Polanski’s official output – for example the much-watched videos of him talking to voters – are for obvious reasons created by professionals. But on X, Bluesky and Facebook, the Green leader likes or reposts content at a dizzying pace.

As such, few were hugely surprised when Polanski pressed the “collaborate” button on an Instagram post comprising about 20 photos from an event he attended, the first a photo of himself.

Near the end of the collection – and never, it seems, properly scrutinised – was a photo of a man wearing a T-shirt showing a guillotine and words from a 1979 song by the band XTC: “We’re only making plans for Nigel.”

The collection was swiftly removed from Polanski’s feed with the party saying the T-shirt was “clearly in poor taste”. Reform have asked the police to investigate.

There was, however, some debate within the party over whether it had been right not to apologise to Farage, the Guardian understands.

That decision drew significant scrutiny overnight after Polanski’s appearance on the BBC’s Newsnight on Tuesday, in which he condemned political violence but refused to say he was sorry three times, insisting that he would not apologise for an “inadvertent mistake”.

It has also left the Greens wondering whether the sort of carpet-bombing approach to social media that helped Polanski build his profile needs to change, and particularly whether he needs help on this.

“There are discussions at all levels about how to avoid this sort of thing happening in the future,” the senior party figure said. “There does seem to be a keenness from people around Zack for him to hand over the keys to his social media accounts.”

Another party source said Polanski had to be rescued from himself when it came to social media: “There is a problem among some people of being terminally online, and Zack is terminally online.”

Not everyone agrees. “The whole point about Zack is that he does his own stuff, and that’s quite exciting for a party leader,” another senior figure argued.

But as he approaches a year in the job, the coming weeks do feel like a moment for Polanski and those around him to consider a stock-take and the next steps for a party that has expanded hugely in membership and relevance since he became leader.

The heady days of the stunning byelection win in Gorton and Denton are now five months in the past and Friday will bring the result of the mayoral byelection in Greater Manchester triggered by Burnham’s departure. Polls suggest the Greens will come third, a big improvement on their fifth place in 2024, but not a moment for renewed euphoria.

Some in the party, while still supportive of Polanski, argue that he needs to find a way to move beyond campaigning mode and begin to look more like someone who could play a part in government.

There is some envy for the narrative created by the communications team around Burnham, albeit with the obvious advantage that they are working with someone who is now prime minister.

“They introduced him very cleverly as this friendly guy in a T-shirt, but it evolved, and now he’s a serious man in a suit meeting world leaders,” one Green source said. “Our videos to introduce Zack were great, but something seems to have gone missing since then.

“The risk is that Burnham ends up looking like a safe pair of hands. If he manages to seem totally sensible, and we seem unsensible, that’s when people might move to Labour.”