On the wide streets around Leeds’ Roundhay Park, Labour canvassers have built up a considerable step count just to walk between each of the stone-built mansions in one of the city’s most affluent suburbs.

Lucy Powell, Labour’s deputy leader, is with activists in the sunshine admiring the manicured lawns and window-box pansies. This is one of the safest wards for Labour in Leeds, with graduates, doctors, lecturers and small business owners.

In years gone by, voters in these houses with sweeping driveways and climbing roses would have been natural Conservatives. A short walk away is Roundhay school, the alma mater of Liz Truss, a place she amusingly tried to paint as the wrong side of the tracks.

But now Labour councillors in the ward have a different adversary – the Green party. It is in these wealthy areas, as well as the more diverse suburbs and student population, that the Greens are making gains.

“There’s no question these are going to be difficult elections,” Powell said as she stops at the church’s community cafe. “Look at where we are in the polls. We haven’t given a strong enough account of ourselves and what we’re doing in government nationally, how we are reshaping the country. People are wanting change and they’re looking elsewhere to vote for change.”

Labour councillor Jordan Bowden has been talking up the Labour council’s own green policies – it was named green council of the year. He talks to residents about solar panels and heat pumps for schools, sewage problems in the waterways around Roundhay Park, bike lanes and electric bikes – one of which Tracy Brabin, the mayor of West Yorkshire, had used to cycle into town.

They had briefly considered calling them “Brabin bikes” – they joked – but unlike Boris Johnson, who took ownership of the cycle scheme his Labour predecessor had introduced, the party are still squeamish about taking credit for their own achievements.

Zack Polanski, the Green party leader, has made Leeds a key target, especially younger, mixed areas like Hyde Park and Armley. Students hosted a fundraising rave for the Greens, attended by Polanski, raising more than £16,000 for the local party campaign. But Leeds is a microcosm of the challenges Labour is facing everywhere – in East Leeds there is a strong Reform presence also targeting gains.

In Leeds, the local Green candidate is another former pupil of Roundhay school, Brannoc Stevenson. Several locals mention fondly they know his family. There is a mix of Green and Labour boards parked on top of the manicured hedges.

Political divides are very unpredictable. Jill, a local homeowner chatting to Powell in the sunshine, said she would consider voting Labour, having previously been a Conservative. She was unconvinced by the Green party. But in the Clocktower Community cafe where Powell did the rounds of the tables over tea and cake, many said they were considering voting Green.

“We’re very proud of the green space around here, I think we could do with someone who will protect it,” one said.

The Green and Labour leaflets bear a striking similarity – independent local bus networks which Labour had legislated to allow, approval for a Leeds tram, which had been promised during the last budget.

It is frustrating, Powell said, to see a party claim credit for policies which have been legislated for by a Labour government.

“It’s being clearer about who we’re not for and who we are prepared to take on, whether that is the social media companies, whether that is the water companies, whether that’s the energy companies who are perhaps profiteering, when everybody else is seeing their bills going up,” she said.

“I think it’s about being a lot clearer that we are prepared to take on these things more strongly, and not cede the progressive ground either. I saw the Greens are trying to take credit for votes at 16. That’s a Labour manifesto commitment, a Labour bill. It’s been a long standing Labour cause to ensure that young people are right at the centre of our politics and our decision making, bringing in votes to succeed.

“We weren’t shouting about it. And if you cede the ground, then others come and play and claim credit for your progressive agenda and what you’re actually delivering. I feel like we’ve let them come on to our ground, rather than they’ve won it over.”

Kathleen Johnstone, a former foster carer and longtime community activist, is standing for Labour in the ward for the first time. The ward extends down to the border of Harehills, one of the poorest pockets of the city, and Oakwood, popular with young city workers.

“I think they see the Greens as a breath of fresh air,” she said. “But I’m not the same-old Labour, I’m me. I want to show what the city council has done, under really hard circumstances. We are finally getting more money from local government, we don’t want the rug pulled from under us.”

Powell said that she remains convinced that many of the voters who are frustrated with Labour – and looking to other parties – can be won back.

“On Gorton and Denton, people were making their minds up right until the last minute. And I think that is true here. There’s people who have made their decision. They’re voting Reform, they’re voting Green, or they’re voting Labour.

“But there’s a lot of soft voters in the middle, to be honest, and it doesn’t ever come through in polls. And there is absolutely an anti-Reform coalition. That is not just the demographic you might imagine. It’s not just Guardian readers. It’s white, working-class people who traditionally vote Conservative or traditionally vote Liberal Democrats. They know the stakes are very high.”