Barnaby Joyce has issued a pointed reminder that the Coalition won the 2016 election due to the National party’s performance, and attacked Liberals for backgrounding against the junior Coalition party.

Joyce has also defended the Liberal Hollie Hughes’s right to inherit the seat of his former deputy, Fiona Nash, but warned there “are other ways to skin a cat”, suggesting the Nationals could ask for more Senate spots in future to compensate for the loss of Nash’s seat.

The Nationals have been criticised in the media for harming the government by failing to check their candidates were not foreign citizens.

On Friday the high court found that Joyce and Nash were ineligible to sit in parliament because of their dual citizenship, forcing Joyce to a byelection in New England and creating a vacancy in Nash’s seat that is likely to be filled by Hughes.

Asked about a report in the Australian in which he reminded the Liberals the Coalition would have lost last year’s election but for the Nationals, Joyce told Radio National on Monday: “I just get annoyed when people, off the record … start making comments about the National party.

“We won the last election because the National party didn’t go backwards, we actually went forwards,” he said. “We held all of our seats and we won one. That’s a statement of fact and people know it. I just think that people should be reminded of that. And if they really have a strong view about something, put your name to it.”

On Sunday Guardian Australia reported that the Nash camp now accepted her political career was unlikely to continue and she was extremely unlikely to make a comeback by seeking re-election.

Asked whether Hughes should step aside for Nash at the next election, Joyce said: “I’m not going to start telling someone who’s been duly elected what they can and can’t do.”

Joyce said the Senate ticket “is what it was” and the process of recounting the votes in the 2016 election ordered by the court would now take place.

He said if the replacement were chosen through a byelection he would “be lobbying as intensely as possible to get Fiona Nash back in” but, as that was not the case, every person on the Senate ticket would “go up a level”.

“There are other ways to skin a cat and we’ll look at them down the track,” he said.

Asked why he had spruiked the credentials of the interim Nationals leader, Nigel Scullion, to be acting prime minister instead of Julie Bishop, Joyce said “of course” he had done so, because Scullion sits in the National party.

Bishop, speaking to the media in Perth on Monday, brushed off suggestions of friction between the Coalition parties, which she described as a “political family” that was the most powerful political force in Australian history.

“There will be issues from time to time but, like any family, you get over them and you move on,” she said.

“So we’re working closely together and our focus at present is to ensure that Barnaby Joyce wins the byelection in New England on December 2, and I’m sure that’s his focus as well.”

Joyce’s task has been made easier by a decision on Monday by the Shooters, Fishers and Farmers party not to field a candidate in the byelection, and One Nation is reportedly also going to stay out of the contest.

Joyce reiterated his suggestion that section 44 of the constitution, which bars foreign citizens, could be reformed at a “bulk” referendum on election day asking Australians about a series of constitutional changes.

On Sunday the attorney general, George Brandis, appeared to rule out change to section 44, contradicting the prime minister by saying the government was “not talking” about a referendum but rather changes to the citizenship laws to address the problem of people being rendered ineligible to stand for parliament.

Joyce said he was not at odds with the government’s position, joking that he has got “the joy” of being “less employed”.

“I’m a candidate for New England; I am Barnaby. And that is great. I’m not the deputy prime minister, I’m not a minister, I’m not in cabinet.”

Joyce said there were many suggested constitutional reforms to deal with, citing Labor’s support for a referendum on constitutional recognition of Indigenous Australians.

Joyce said the “overwhelming sentiment” in New England was to question how it was possible that he was born in Tamworth and was “somehow not an Australian”.

The high court unanimously held that Joyce held New Zealand citizenship by descent from his father, who was born in Dunedin.

The justices said while it “may be said that it is harsh” to disqualify a candidate born in Australia who considers himself or herself exclusively Australian, the nomination process requires candidates to declare they are eligible for election, which was an “occasion for serious reflection”.

They also noted that references to the court only arise “where the facts which establish the disqualification” have been brought to light, and a candidate “need show no greater diligence” discovering those facts than the person who brought them to the attention of parliament.