Islington. It’s the idea, as much as the place itself, that the right hates: an enclave of wealthy people who have the temerity to vote against right-wing interests. The real Islington, and Jeremy Corbyn’s patch of it in particular, is not all like that. Although parts of his constituency do resemble the cliché of large townhouses and overpriced flat whites, Labour’s 78-year hold on the seat is founded not on the palatial houses around Highgate Hill but on the constituency’s many council estates.

It’s a place I know well. As a child, Islington North was the place next to the edge of the known world, or, as I would come to call it later in life, Barnet. After going to church in Bow, my mum and I would take the bus through it to choir practice, where I sang until my voice broke, in both senses of the word.

Today, austerity is making Islington North look more like its past. Not the Islington of my teenage years, but of my childhood: grimy streets and growing homelessness. Outside the Archway McDonald’s an elderly woman points out the evidence of last night’s clubbers and tells me that today’s teenagers are less considerate than I was or her grandson is. She’s wrong; I once vomited in that same street. But street-sweeping, particularly at night, has been one of the first things that councils have cut back on under constraints from decreasing local authority budgets.

As for homelessness, that, too, has come full circle. Tony Blair’s government was the first to count the number of people sleeping rough, and by the time Labour left office it had been reduced by two-thirds. In the six years since David Cameron first came to office, the homeless figure in England more than doubled from 1,768 estimated rough sleepers to more than 3,569 today. This is the world that Jeremy Corbyn’s supporters want to fight against. These are the effects of Conservative rule that make Labour activists yearn for an anti-austerity champion.

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Demolishing the stereotypical views of Islington and elsewhere is vital if we are to understand the currents flowing through ­Labour. This summer, there have been three main characters in the soap opera (or farce) that has played out in the party – the beleaguered leader, Jeremy Corbyn, of Islington North; the leading rebel, Angela Eagle, whose constituency is in Wallasey; and finally, the eventual challenger, Owen Smith of Pontypridd. I visited all their constituencies in a whirlwind week in the hope that it would illuminate the leadership race and the wider challenges for left-wing politics in Britain.

In all three places, the easy assumptions about Corbyn’s appeal were complicated by the facts on the ground, but a common thread united them. Outside the Holloway Road Odeon, I heard it first: “Jeremy is a nice guy, but he’s not a leader.” The trouble was that even those who questioned Corbyn’s leadership had little faith in those challenging him.

On 4 July, during a meeting of the Parliamentary Labour Party, Neil Kinnock talked about “the supermarket test”: how people in Tesco or Lidl would say “I want to vote Labour, but I can’t vote for Ed Miliband”. He urged Labour’s representatives in the Houses of Parliament to “apply the supermarket test for Jeremy Corbyn and see what answer you get”.

In reality, they had been applying it for months. That was the spur to the attempts in late June to oust Corbyn as Labour leader. For the 172 MPs who said they had no confidence in him – and the 41 per cent of Labour members who told YouGov that they thought Corbyn was doing either “fairly badly” or “very badly” – he is an obstacle on the road to saving Britain from the Tories. Idealism didn’t create a minimum wage, set up Sure Start centres, or bring in civil partnerships: assembling a broad enough coalition to elect a Labour government did.

The minority of MPs who support him, and the thousands of members who say they will vote for him again, feel differently. For them, Corbyn’s demise would feel like a capitulation. It would feel like ­accepting that neoliberalism, capitalism and austerity have won the day, that the role of the Labour Party is to ameliorate rather than oppose them.

When I visited Islington North, Labour’s leadership election was only just starting to get under way and Angela Eagle was still in contention. Her tough performances deputising for the leader at PMQs have made her popular at Westminster but that enthusiasm has not made it as far north as Islington. “To me, I can’t see Angela Eagle as a prime minister either,” said Mike, one of the regulars at the Coronet, a Wetherspoons on the Holloway Road. “What are they running her for?”

The same sentiment prevailed in Wallasey, the Wirral constituency that Eagle has represented since 1992. There, too, were a few pockets of Corbynmania. There was also a sense that Labour is heading for defeat as long as Corbyn remains in place – but little faith in Eagle’s ability to alter that trajectory.

Wallasey is of less long-standing Labour vintage than Islington North. It remained steadfastly Conservative even in the landslide years of 1945 and 1966, with Eagle the first Labour politician to win the seat in 1992. Although she is now in possession of a 16,000-vote majority, her neighbour Margaret Greenwood took Wirral West seat back from the Conservatives by a margin of only 400 votes. Tory strategists still eye the Wirral hungrily.

Wallasey is home to New Brighton, the seaside resort commemorated in Martin Parr’s 1985 series The Last Resort. A popular tourist destination for most of the first half of the 20th century, New Brighton was hurt by tidal changes in the River Mersey, which stripped most of its sand, and by the closure of its pier, but it remains a favoured destination for retirees and day trippers. In times past, Liverpool families that did well for themselves crossed the Mersey, bought a home – and promptly started to vote Tory. Wallasey, and the Wirral as a whole, is still where Scousers who have made it good set up their homes, but nowadays their politics usually survives the river crossing unscathed.

Yet there is still a vestigial sympathy for Conservatism in the leafier parts of Victoria Road and Seabank Road, one that is largely absent from Islington North. Perhaps Theresa May’s diligence in dealing with families affected by the Hillsborough disaster, which was mentioned frequently when I asked people for their opinion of the new Prime Minister, is sufficiently well regarded here that it is beginning to erode the Thatcherite taint still hanging over the Tory rosette on Merseyside.

However, it is not just Labour politics that is proving increasingly capable of weathering the journey across the Mersey. In Westminster, the chatter is that Militant – driven out of Labour in the 1980s, though most of its members continued to live and work on Merseyside – is back as a force in the city’s constituencies, and that many of its members have moved out and retired to New Brighton. Their influence is blamed for the series of damaging stories that slipped out of Wallasey in the days after Eagle declared her candidacy.

“There’s a reason why they’re so good at getting themselves on the national news and in the papers,” one MP tells me. “It’s that they’ve done all this before.”

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The perception that Eagle “lost control” of her local party, as well as a disastrous campaign launch, led to support from fellow MPs ebbing away from her. It went instead to Owen Smith, the MP for Pontypridd, a little-known figure outside Westminster, but one who has long been talked of as a possible Labour leader inside it.

Smith’s great strength, at least according to some of his backers, is that he is a blank canvas. Certainly, as with Corbyn in Islington, there was a widespread perception in Wallasey that Eagle was not cast from the material from which leaders are made. Smith at least had the advantage of introducing himself to voters on his own terms.

His slim hopes of defeating Corbyn rest on two planks. First, the idea that a fresh face might yet convince wavering members that he could win a general election. A vote for him rather than Corbyn can therefore be seen as a vote against the Conservatives. Second, he is willing to call for a second European referendum. Among Labour Party activists, who backed staying in the European Union by 90/10 per cent, that is a compelling offer.

In Islington and Wallasey, both of which voted Remain (and both of which still have houses flying the flag of the European Union when I visit), that message also has wider appeal. But in Smith’s own seat, a second referendum is a tougher sell. The Valleys voted to leave by a near-identical margin to the country at large. No one to whom I spoke was enthused about replaying the referendum.

Smith’s status as a “blank slate” will only be useful if he manages to write something appealing on it over the course of this summer. It is also possible he could just remain largely unknown and undefined.

Travelling around the country, I became accustomed to explaining who he is. Even at my hotel in Cardiff, which borders his constituency, the name “Owen Smith” was met with blank looks.

Unfortunately, the habit proved hard to break once I was in Pontypridd, resulting in an awkward scene in the back of a taxi. “I know who my MP is,” my driver said angrily, before launching into a lengthy diatribe about the arrogance of London-based journalists and a London-led Labour Party. The accent had changed, the setting was more confrontational, but the story remained the same as in Islington and Wallasey: he was convinced of neither Jeremy Corbyn’s nor Angela Eagle’s ability to fight and win an election. “That voice? In a room with Putin?” he said of Eagle. Then he said something unexpected. “But I’ll tell you what – they need a change from Jeremy Corbyn – and why not Owen Smith?”

“Why not Owen Smith?” As much as they might wish to deny it, that is the message with which Corbyn’s critics will try to take back control of the Labour Party. It is a message that feels unlikely to move or inspire. As I catch the train back to London, I reflect that those who want to convince Labour activists to give up Jeremy Corbyn – and what they feel he represents – need to offer them something compelling in return. No one puts “Vote for the lesser of two evils” on a banner.