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Liz Truss’s bid to become prime minister has received a much-needed boost after leading Eurosceptics, including the attorney-general Suella Braverman, backed her campaign.

Truss also won the support of Braverman-backer Steve Baker, the former Brexit minister, and Lord David Frost, who delivered a scathing attack on Truss’s rival Penny Mordaunt, the trade minister, in the process.

The foreign secretary is the favoured candidate of supporters of Prime Minister Boris Johnson and has sought to appeal to the small-state, low-tax, pro-Brexit wing of the Conservative party.

But she has been struggling to build momentum in her bid to become the next Tory leader. Mordaunt has been a surprise success in the leadership campaign, with a message combining Brexit, social liberalism and the offer of a “fresh start”.

Truss remains in third place behind former chancellor Rishi Sunak and Mordaunt, making the next few days crucial for her campaign as she tries to gain ground on her rivals.

Braverman, who was eliminated from the leadership contest in the second round of voting by Tory MPs on Thursday, announced she would back Truss to deliver on Brexit and to take a firm line on immigration, cutting taxes and shrinking the state. Braverman, who described herself as the only “authentic” Brexiter, secured only 27 votes.

Truss won 64 votes in the second round, trailing Mordaunt on 83 and Sunak on 101. She will hope to pick up many of Braverman’s 27 votes in the next round of voting on Monday.

In his Daily Telegraph column, Frost urged another pro-Brexit candidate, Kemi Badenoch, to stand down “in return for a serious job in a Truss administration” to allow the Tory right to unite.

Badenoch, who won 49 votes on Thursday, said she had no intention of withdrawing from the race and was in it “to win it”.

Frost also renewed his criticism of Mordaunt, with whom he worked on Brexit issues as a cabinet office minister. He has said she “wasn’t fully accountable or always visible” during talks with Brussels.

Badenoch, who also has gained momentum in the contest, and Tom Tugendhat, the moderate One Nation candidate who won 32 votes, hope to use a series of television hustings to boost their profile.

Channel 4 will host a debate on Friday night, with ITV hosting a second event on Sunday and Sky News early next week.

If Tugendhat is knocked out in the next round of voting on Monday, Sunak will hope to pick up many of his votes; he needs the backing of 120 colleagues to be sure of reaching the final shortlist of two, which will be presented to party members over the summer.

If Badenoch is eliminated next, then Truss and Mordaunt will be locked in a desperate tussle to win the backing of her supporters to reach the final run-off. A new Tory leader and prime minister will be named on September 5.