The Conservative party will not win the next general election if it responds to the rise of Nigel Farage’s Brexit party by shifting dramatically to the right, leadership contender Matt Hancock warned on Sunday.

In an interview with the Observer, the 40-year-old health secretary says the way to defeat Jeremy Corbyn and Farage is to deliver Brexit with a deal, then dominate the centre ground by focusing on improving public services and delivering prosperity for all, including by increasing the living wage to more than £10 an hour by 2022.

His call reflects mounting anxiety among moderate Tories that their party will react to Farage’s success in the European elections and its near miss in last Thursday’s Peterborough by-election when it narrowly failed to win its first seat in parliament, by electing a new leader committed to leaving the EU with no deal and a rightwing agenda of tax cuts.

By contrast, Hancock insisted that the challenge was to win over a cross-section of voters, including more young people, women and those from ethnic minorities. “We cannot be shifted off the centre ground by the Brexit party,” he said. “We can’t just occupy the centre ground. We need to dominate [it]. We have got to win back voters both from the Brexit party and those who voted Lib Dem, as well as people who may have considered voting for Corbyn. If we only seek voters from the Brexit party, it is hard to see how we would win a general election.”

In a pitch to centre-ground Tories before MPs vote for the first time on Thursday on who they want to succeed Theresa May, Hancock said that the next leader had to be upfront with the British people about the “realities and trade-offs” of leaving the EU and accept that parliament would not agree to leaving without a deal.

Once Brexit had been delivered, his party had then to win the battle of ideas with Labour by showing the benefits that free enterprise can deliver through improving public services and ensuring everyone benefited from economic success by boosting wages. “The strong economy means there is fiscal firepower and we need to use that to show people the benefits of the free-market system. The big challenge of the 2020s is to win the fight for free enterprise, which is under threat as never before in my lifetime.

“One of the ways we do that is use economic growth and using that fiscal firepower to deliver high-quality, properly funded public services and to make sure that people have better living standards.” He added: “The priority has to be putting forward plans that are properly costed and tax cuts when we can afford them.”

Several other Tory leadership contenders, including Dominic Raab, Sajid Javid, Esther McVey and Jeremy Hunt, have said they would make a priority of tax cuts if they become PM. Raab has pledged to cut the starting rate of income tax from 20% to 15%.

Concerns about rash promises on tax were also voiced by former Tory minister Lord Willetts, now president of the independent thinktank The Resolution Foundation. Willetts said: “The leadership contest is an opportunity for the party to advance its thinking as the world around it changes. Tax cuts have been part of the Conservative offer but now things are different. Admiring Ronald Reagan and Margaret Thatcher doesn’t mean it’s the 1980s. For a start, the financial crash and deep recession have doubled the national debt from 40% to 80% of GDP. We have worked hard to bring down public borrowing from 11% to 1% of GDP. We should not reverse that progress just to finance tax cuts.

“So any tax cuts have to be matched by cuts in public spending. But we are coming to the end of a benign period when the big generation of Boomers were of working age, paying taxes, and not heavy users of public services. Now [they are] growing older.

“The Resolution Foundation estimates they are pushing up public spending on the NHS and pensions by £36bn over the next decade. So we need to start working out the best way to pay for this. Of course, more economic growth would help, but that would not just follow from tax cuts. It requires tough decisions to invest more in infrastructure, ease planning rules which are a barrier to development, and investing in the technologies of the future. And some of these options would have to be financed out of higher taxes as well.

“So a serious leadership contest would be about how we pay for Britain’s fiscal and demographic pressures. The candidates may spar over tax cuts, but as PM they’ll more likely wrestle over tax rises.”

Michael Gove sought to focus once more on policy issues after the revelations about his past cocaine use and, writing in the Sunday Telegraph, he set out a “pro-business economic plan” to take on Corbyn’s “Marxist message”. He said his plan would mean “reducing the regulations which hold business back”, cutting taxes and “using the opportunity of life outside the EU to look to replace VAT with a lower, simpler sales tax”.

Writing in Sunday’s Observer, shadow chancellor John McDonnell attacks the Tory leadership candidates, saying that their “obsession with slashing taxes is just the latest evidence of the intellectual bankruptcy of the right, which is now widely apparent even to some Conservatives”.

McDonnell says Labour will increase taxes for the top 5% of earners and clamp down on tax dodging. “Only Labour will tax the rich and giant corporations to end austerity, fund our public services properly, and rebuild our economy so it works for the many, not the few.”