Boris Johnson has said he will win a new Brexit deal at next month’s EU summit by using his “powers of persuasion”, and rejected calls for a further extension to article 50.

The prime minister said he had no plans to accept new legislation that would require him to write to the EU asking for a “pointless” delay to Brexit.

“We’ve spent a long time trying to sort of fudge this thing and I think the British public really want us to get out. They don’t want more dither and delay,” he said during a visit on Friday to a farm near Banchory, Aberdeenshire, at which he encountered a prize bull called Keene.

Asked how he would secure a new deal at the EU summit on 17 October, he said: “By powers of persuasion. Because there’s absolutely no doubt we should come out … It’s a pointless delay.”

Johnson, who gave hesitant and incomplete answers to questions about the crisis at Westminster, was pressed by reporters on whether he would ignore a legal requirement to ask for a delay.

He cut short one part of his answer and said he did not want such a delay. “I will not ex… I don’t want a delay,” he said.

The prime minister was in Aberdeenshire to announce more than £50m in extra funding for Scottish farming following a government review of agricultural support, and another £160m in backdated funding.

He had earlier met fishing industry leaders at Peterhead fish market on Friday morning, and is later on Friday due to meet about 100 Conservative party members in West Aberdeenshire and Kincardine, the seat held by the Tory MP Andrew Bowie.

Johnson was asked twice whether he would sack his chief adviser, Dominic Cummings, after the former prime minister Sir John Major described him as a “poison” at the heart of government on Thursday night.

Johnson did not answer the first question on his adviser’s future but pressed a second time refused to give Cummings his unambiguous support. He answered: “Look… advisers, as I think someone said in the Commons the other day, advisers advise and ministers decide.”

He was also asked about his own future after a tumultuous week during which he suspended 21 Tory MPs, including the former chancellor Ken Clarke, his younger brother Jo Johnson stepped down from the government and as an MP, and other senior Tories announced their retirement. His brother cited irreconcilable conflicts between his family and the national interest.

Asked about his failure to uphold his pledge during the Tory leadership campaign to unite the party, Johnson said he had promised to “deliver Brexit, unite the country and defeat Jeremy Corbyn. And that’s what we’re going to do.” Asked again when he would resign, he said: “Er … Well … I think after those three objectives have been accomplished I will … At some point after those three objectives have been accomplished.”

He said he was “perplexed” that neither Jeremy Corbyn or the SNP would agree to a snap election on his timetable. “I have never known an opposition in the history of democracy that has refused to have an election,” he said.

“I think they obviously don’t trust the people. They don’t think that the people will vote for them, so they’re refusing to have an election. So what we will do is we will go to the summit on the 17th. We will get a deal and go out from 31 October.”

He did not deny the SNP could win more Westminster seats in a snap election but ruled out giving Nicola Sturgeon, the first minister, the legal powers to stage a fresh independence referendum which she plans to ask for before Christmas.

“People were told in 2014 that the referendum was a once in a generation event. I don’t see why they should go back on it,” he said.

Johnson is due to stay with the Queen at Balmoral on Friday night with his partner, Carrie Symonds, in line with a tradition for prime ministers during annual summer break there. The Mail on Sunday reported last month that Symonds and Johnson would share a room, despite a long-standing convention where unmarried couples are barred from doing so.