Mawson began karting at the age of seven and quickly dominated the junior karting scene in Australia, winning nineteen state titles and three national titles between 2003 and 2012. He finished second in the Junior Max category at the 2011 Rotax World Championships and was named Australia's best karter in 2010 by various media outlets.
Mawson made his open-wheel debut in French F4 in 2014, qualifying on pole position and winning on his debut at the inaugural round at the Bugatti Circuit. He completed all but the final round of the season, finishing fourth overall with three wins and 188 points.
In 2015, he moved to ADAC Formula 4 with Van Amersfoort Racing, competing alongside Harrison Newey and Mick Schumacher. Despite the pressure from two highly rated teammates, Mawson outperformed them both, finishing third in the championship with five wins and missing second place by just two points.
He remained with Van Amersfoort Racing for 2016 and dominated the ADAC Formula 4 season, winning ten of the twenty-four races to claim the championship by 52 points from Schumacher, who had since moved to the Prema team. As a reward, Mawson was given the opportunity to test one of Van Amersfoort's European Formula 3 cars at Monza, where he was able to trade fastest laps with Lando Norris.
In 2017, Mawson graduated to European Formula 3 in a continuing collaboration with Van Amersfoort Racing. The season proved difficult; he finished only thirteenth in the standings with a single podium finish at the Nürburgring. In 2018, he contested the final season of the GP3 Series with Arden International, finishing thirteenth with two podiums — one in France and one in Russia.
Mawson then switched to the Porsche ladder in 2019, competing in the Porsche Supercup with Team Australia and also contesting a handful of Porsche Carrera Cup Germany races at the end of the year ahead of a full campaign in 2020.
Mawson returned to Australia in 2021 as a late entry to the S5000 Championship. He completed the season with three race wins to claim the title ahead of Thomas Randle, then successfully defended his championship in 2022.
In June 2023, midway through the 2023 S5000 season, Mawson was provisionally suspended by Motorsport Australia after failing a doping test. In September 2024 he was issued with a three-year competition ban backdated to May 2023 for an anti-doping rule violation, having tested positive for meldonium contained within a supplement he had used.
Mawson's 2016 ADAC Formula 4 title — achieved by defeating a field that included Mick Schumacher — remains the high point of his open-wheel career and marked him as one of the strongest junior drivers of his generation. His subsequent return to Australia demonstrated sustained competitiveness at national level, though his career was interrupted by the doping ban that ended his involvement in motorsport for three years.