The Centre Alliance’s Rebekha Sharkie, who quit parliament in the citizenship crisis, has an early lead in her fight to win back her South Australian seat of Mayo, according to a ReachTEL poll.

Sharkie is ahead of her opponent, the high-profile Liberal Georgina Downer, 58% to 42% on a two-party basis.

The poll was commissioned by the Australia Institute, a progressive think tank.

The Liberals have been hopeful of capturing Mayo, thus increasing their parliamentary majority. They held the seat until Sharkie, as part of the Nick Xenophon Team (since renamed the Centre Alliance) won it in 2016 from Jamie Briggs, who earlier had to resign from the ministry over a personal indiscretion.

Georgina Downer is the daughter of Alexander Downer, a former occupant of the seat who was foreign minister in the Howard government.

The vote is on July 28, when five byelections will be held in a Super Saturday across the country. The Liberals believe the long campaign will favour them in Mayo because Sharkie has much less in terms of resources. She is, however, well known in the electorate.

The poll has her on a 40.1% primary vote, with Downer on 34.4%. Labor is polling 7.7%; the Greens 10.7%. The vote for “other/independent” is 3.5%, with 3.6% undecided. The sample was 1031 with polling on the night of June 5.

The poll also asked whether company tax for large companies like banks, mining companies and supermarkets should be increased, kept the same or decreased. The results were: increased, 25.4%; kept the same, 44.9%; decreased, 24.8%.

People were overwhelmingly opposed to the banks receiving a company tax cut (69.1% against).

Asked how they would prefer to spend the $80 billion proposed to be spent on company tax cuts, 51.4% chose “infrastructure and government services like health and education”, 6.5% said personal income tax cuts, 29.1% said decreasing the deficit and repaying debt, and 8.7% said proceeding with the company tax cuts.

Asked how the Senate should vote on the third stage of the budget’s income tax policy, which “removes the 37-cent income tax bracket altogether, meaning someone earning $41.000 would pay the same marginal tax rate as someone earning $200,000”, about two thirds (65.3%) said the Senate should oppose it, while 25.2% said it should vote in support.