The lush hills and fields of Brecon and Radnorshire, where sheep outnumber people by 10 to one, are fertile ground for a Liberal Democrat revival. Yet in this deeply split constituency, which voted 52% leave, there is no “bollocks to Brexit” on any Lib Dem leaflet, nor any mention of leaving the EU at all.

It may seem odd to omit what has become the Lib Dems’ defining selling point from the literature of its candidate, Jane Dodds, though in between pledging investment in health and social care and delivering for farmers, she promises to “fix our broken and divided politics” – code for disillusion with the major parties and the failure of Westminster to break the Brexit impasse.

Dodds is cautious in the way she talks to voters about Brexit, restricting herself to discussing the perils of no deal and its effect on the constituency’s sheep farming and automotive workers, though she says the party’s stance is hardly a secret.

Unlike many MPs, and some of the activists who have come to the constituency to campaign, hoping to help stop Brexit, she says the 2016 referendum made her resolve to listen more.

“The referendum result was a shock, it made me think more than anything though that we need to get out more and talk to more people,” says Dodds, the leader of the Welsh Liberal Democrats.

Dodds is keen to keep to local issues on the doorstep. Photograph: Gareth Phillips/The Guardian

In Brecon and Radnorshire, she says she has found a deep disconnect between its inhabitants and Westminster politics, often a physical one because of a loss of local services and frustration with issues such as broadband infrastructure. No deal will “potentially decimate” sheep farming, the economic and cultural lifeblood of many communities, she says.

“The first thing that I’m clear about is that a no-deal Brexit must be taken off the table,” says Dodds, drinking coffee in the sunshine in Crickhowell before a day of canvassing. “But beyond that, people do know what the Lib Dem position is, though – we aren’t dishonest about that – and they know we are a remain party who will campaign for a ‘people’s vote’.”

Though Dodds is keen to keep issues local, the vote on Thursday will have huge national significance. Chris Davies, the ousted Conservative MP convicted of expenses fraud, is fighting again for the seat after a byelection was triggered by a recall petition.

Quick Guide Brecon and Radnorshire byelection Show Why was there a byelection? A petition to recall the Conservative MP Chris Davies secured the signatures of more than 10% of registered voters in the constituency. Davies was convicted for submitting fake expenses documents invoices for nine framed landscape photographs, costing £700, to decorate his office in Builth Wells. He was fined £1,500 and ordered to complete 50 hours of community service. He stood again for re-election. How did the parties do at the last election? In 2017 the results were as follows: Christopher Davies (Cons) 20,081 (Maj 8,038)

James Gibson-Watt (Lib Dem) 12,043

Dan Lodge (Lab) 7,335

Kate Heneghan (Plaid) 1,299

Peter Gilbert (Ukip) 576 What were the 2019 results? The Liberal Democrats took the seat from the Conservatives: Jane Dodds (Lib Dem) 13,826 (Maj 1,425)

Christopher Davies (Cons) 12,401

Des Parkinson (Brexit) 3,331

Tom Davies (Lab) 1,680

Lady Lily the Pink (Monster Raving Loony) 334

Liz Phillips (Ukip) 242

Should Dodds win, the Conservative majority will dwindle to just one seat, just a week after Boris Johnson was installed as prime minister. Conservative campaigning has been low key, though Johnson is scheduled to visit on Tuesday. There are some nerves about a “Boris bounce”, but Lib Dems are outwardly confident of taking the seat, believing that Davies’ reselection as a candidate is a sign that CCHQ wants a convenient excuse for a loss. The Brexit party will also stand, making it more likely the leave vote will split.

Nigel Dances, 62, a project engineer in the car industry, is a particularly receptive punter on the doorstep. For him, a no-deal Brexit would be personal, though he is wary about stopping Brexit entirely. “I’ve always believed it was better to stay in and particularly for my business it would be catastrophic if we had no deal,” he tells Dodds. “The extra taxes make a huge difference to the margins of the business. But I do believe we need to respect democracy and people voted to come out. I don’t know if they ever realised the implications, that’s the thing.”

Local Lib Dem campaigners know the party’s positioning on Brexit will mean some voters are out of reach. Though 19% of voters signed Davies’ recall petition – almost double what was required – many will have been angry leave voters.

Antonia Harrison, an activist who travelled from Portsmouth to campaign, has had to temper her pro-European enthusiasm when speaking to voters. “I’m a European first and foremost: it’s my identity, it made me determined to try and stop this,” she says. “But it is different here, I find a lot of people are saying they voted to leave. Some have said they would vote Lib Dem but because they voted leave, then they can’t this time.”

One of the biggest challenges is getting around a constituency the size of Luxembourg mostly crossed by winding country lanes. Dodds says she is walking about 20km a day and driving between districts as much as three hours apart. On election day, activists will need to cover a huge area as turnout could make the difference.

The Lib Dems have attracted campaigners from across the UK, as well as two who have flown over from Hong Kong and Toronto. Local organisers claim volunteers include former Tory members from London and current Labour members.

On Crickhowell’s high street, opinion is divided. Robin Masefield, 33, the owner of Natural Weigh, which sells food and household products without packaging, is keen for a Dodds victory. “I hope she gets it, I just don’t support the Conservatives’ political views whatsoever,” he says.

Dodds with Robin Masefield of Natural Weigh. Photograph: Gareth Phillips/The Guardian

In his butcher’s shop a few doors down, Michael Cashell says he believes there could be shy residual support for Davies, especially among Brexiters. “I know people who voted to have him [Davies] deselected and now have his poster up in their window. For people who voted to come out, it is better to have someone who is for Brexit in there,” he says.

“Boris is a character, there’s no doubt about that but I do think he is likely to keep his promises. I think he’s for us: Britain will come first.”

Many residents are amused, more than anything else, that the media circus is now making its way to Brecon. Marianne, a pensioner travelling down Crickhowell high street, said she had been surprised to find the former Lib Dem leader Tim Farron ringing her doorbell. “I couldn’t believe it,” she says. Though it is local issues that will affect her vote, she says many people have found Davies to be personable and effective. “Local people are so fed up about Brexit I think they won’t want to think about it and will vote for whoever they would usually vote for. It is local things that matter more.”