Boris Johnson sought to turn around his scandal-hit election campaign on Wednesday as he gave his first major speech, attacking Jeremy Corbyn for planning tax rises to fund “deranged” nationalisation and Nigel Farage for thriving on Brexit not happening.

The prime minister promised activists in Birmingham that he would get Brexit completed within weeks and spend next year focusing on the NHS, schools and other priorities, as he tried to draw a line under a disastrous few days for the Conservatives.

He attempted to reposition himself as leader of a “moderate, one nation Conservative” government. However, his campaign has already been overshadowed by a series of damaging blunders in recent days. His ally, Jacob Rees-Mogg, suggested that victims of the Grenfell Tower fire lacked common sense; and Alun Cairns, the Welsh secretary, resigned over his knowledge of a former aide’s role in allegedly sabotaging a rape trial.

Johnson is now facing calls to remove Cairns as a Conservative candidate, with the furore threatening to derail the Tory campaign in Wales.

The cabinet crisis erupted just as Johnson went to Buckingham Palace to discuss the dissolution of parliament with the Queen.

He later gave a speech on the steps of Downing Street insisting he had not wanted an election – and claiming he was “chewing my own tie” with frustration that Brexit had not yet happened.

Later, in Birmingham, Johnson was flanked by his remaining cabinet and arrived to chants of “Boris, Boris”, shortly after a group of anti-racism campaigners were manhandled away from the venue by security.

He focused his speech on his intention to pass his “oven-ready” Brexit deal through parliament quickly if he gets a majority and promised a year of “growth and prosperity”.

But Johnson also launched a personal attack on Corbyn, claiming he was a proponent of “Bolivarian socialism” who would pursue a “deranged” programme of nationalisation and bring in higher taxes on “pensions, corporations, inheritance, homes, gardens”. The Labour manifesto has not yet been published and the party has said tax rises would focus on the very wealthy and companies.

He also said Corbyn wanted to have totally unlimited immigration, even though the party has not unveiled its policy in this area yet, and wanted to “expropriate” the assets of schools – referring to a motion at party conference that called for the assets of private schools to be redistributed. This is unlikely to make it into the party’s manifesto.

Johnson did not attack Farage, who is popular with many Brexiters, by name but said those who “cast aspersions” on the deal with the EU have a vested interest in Brexit not happening.

“They are like candle-makers at the dawn of the electric lightbulb,” he said, adding that there was a “terrible sense they are about to lose their market”.

Johnson moved to criticise Farage after the Brexit party leader refused to stand down candidates – despite pleas from pro-leave Tories that by doing so he could prevent a split in the Brexit vote. Farage’s party is still taking a substantial chunk of leave support, which Johnson would need to eat into if he has a chance of winning a majority.

The Conservatives remain ahead in the polls but it is not clear yet if events of the last few days will have affected their chances.

The party’s troubles with the Welsh Conservatives are particularly serious for Johnson as the Tories would need to make inroads in leave-voting areas of the country to win the election outright.

Cairns sent a letter to Johnson on Wednesday saying he was confident he would be cleared by any investigation into his behaviour, but he was stepping down because it was such a “sensitive matter”.

Johnson thanked Cairns for his role in abolishing toll charges on the Severn Bridge and his “unstinting record of service for the party in Wales”.

However, Corbyn said the prime minister should prevent Cairns from standing again to be an MP. “First of all, there’s a victim here,” he told broadcasters after a speech in Telford. “That victim needs to be apologised to and needs to be supported.

“Secondly, if he’s resigning because of his behaviour and involvement in this as the secretary of state, then shouldn’t he also be resigning as a Conservative party candidate in the election? That’s my question.

“Obviously, legally he can stand as a candidate, but does he have a moral right to stand as a candidate? If he’s stepping down as a minister because of his involvement then I would have thought the very least the Conservative party can do is not put him up as a candidate in the next election.”

Keir Starmer, the shadow Brexit secretary and a former director of public prosecutions, said: “He is right to resign but he has got to face an investigation. This is a very, very serious issue. It goes well beyond a ministerial investigation. Resigning was right to do, but it is not enough on its own.”