Liz Truss will seek to reinvigorate her campaign to be the next prime minister by promising an “aspiration nation” on Thursday, after she was unexpectedly beaten into third place by Penny Mordaunt in the first round of voting.

Long seen as one of the favourites to be the next Tory party leader, the foreign secretary has the backing of Boris Johnson super-loyalists Nadine Dorries and Jacob Rees-Mogg.

But she was left trailing in third place after the secret ballot of Conservative MPs, with 50 votes to 67 for Mordaunt, a trade minister and former defence secretary, who had launched her own campaign at a packed event earlier in the day.

Former chancellor Rishi Sunak came first, with 88 votes. That was less of a commanding lead than Johnson had in the early stages of the 2019 contest.

Former health secretary Jeremy Hunt and chancellor Nadhim Zahawi were both knocked out of the contest. Zahawi accepted the job last week, before urging Johnson to resign little more than 24 hours later.

Truss will say in her speech on Thursday: “My mission is to make our country an aspiration nation, where every child, every person has the best opportunity to succeed.”

She will stress her modest background, saying: “I grew up in Paisley and went to comprehensive school in Leeds. I saw children who failed and were let down by low expectations. They will never be let down again on my watch.”

Truss will also reiterate her promise to cut taxes, including by reversing the recent national insurance increase, which is earmarked to pay for social care. In an interview with the Spectator, she suggested tax cuts could be paid for through extra borrowing.

Chair of the independent Office for Budget Responsibility, Richard Hughes, warned on Wednesday of the impact on the treasury’s coffers of the blizzard of tax cuts being promised by Tory leadership candidates.

“A loosening of fiscal policy is not going to improve the sustainability of the public finances. It will make it worse,” he told MPs on the Treasury select committee.

In a YouGov poll of Tory members, which was quickly being widely shared by Mordaunt’s campaignon Wednesday, she emerged as the candidate most likely to beat Sunak if MPs place her in the final two. Asked to rank the two head to head, 67% of grassroots members picked Mordaunt, and just 28% went for the former chancellor.

On Wednesday night, Hunt dashed the hopes of the centrist MP Tom Tugendhat’s team that he would back their candidate, instead issuing a statement saying he would support Sunak.

The former chancellor was “one of the most decent, straight people with the highest standard of integrity that I have ever met in British politics”, Hunt told the BBC. “In a period when we need to rebuild trust with the electorate, that shows we are in tune with modern Britain.”

At her launch event where she was introduced by former business secretary Andrea Leadsom, Mordaunt promised a “relentless focus on cost of living issues”, and claimed to be the candidate that Labour feared most.

Former Brexit secretary David Davis, who is backing her, called it “the best launch I have ever seen”. “The character of the candidate came across,” he added. “None of the other candidates have been willing to give the kind of punchy responses she gave.”

At one point, asked about her definition of a woman, Mordaunt sought to quash claims that she is too “woke” to be Tory leader.

“I think it was Margaret Thatcher who said that every prime minister needs a Willie. A woman like me doesn’t have one,” she said. Thatcher’s famous remark referred to her de facto deputy, Willie Whitelaw.

Davis claimed Mordaunt’s strong showing in the YouGov poll suggested “the party wants a reset”.

Voting in the Tory leadership race came after Johnson dropped a hint that it would be his final prime minister’s questions, saying he was “proud of the leadership I have given and I will be leaving soon with my head held high”.

Downing Street had made tentative enquiries about an international trip for Johnson next week – which would mean he would miss his final outing. But sources said it was now unlikely to go ahead meaning Johnson will face Keir Starmer one last time next week before parliament rises for the summer recess.

Labour and the Lib Dems have complained that the business of government has effectively been paralysed since Johnson announced his intention to resign last week, despite the urgency of the cost of living crisis.

One supporter of Tugendhat, who secured 37 votes on Wednesday, said they were hopeful of winning over at least half of Hunt’s 18 backers and Zahawi’s 25.

They stressed Tugendhat would not drop out of the contest because the main aim was making it to the TV debates – starting on Channel 4 on Friday – so he could “do a David Cameron” and show he was the best candidate to connect with the public.

Meanwhile, Sunak backer Gavin Williamson was accused by one person in the Tugendhat camp of trying to persuade him to throw his weight behind the former chancellor.

Staunch Brexiters Kemi Badenoch and Suella Braverman, who got 40 and 32 votes respectively, insisted they would remain in the race, though there were rumours that one or the other could step aside – perhaps in exchange for a cabinet post from one of the frontrunners.

“I think you’re going to see some movement,” said one Conservative MP. “The pressure will be on Suella, with 32, to fold: she might as well take a job.”

A second round of voting will be held on Thursday, with the bottom candidate eliminated. Another round will then take place on Monday, after two televised debates among the remaining candidates. Once the field has been narrowed down to two, grassroots Conservative members will make the final choice.