Born in the
Czech Republic, Kahun moved to Germany with his family as a child. Kahun played junior hockey with the Jungadler Mannheim U18 team, where he was the league's leading scorer for the 2011–12 season. He then moved to Canada where he played two seasons of major junior hockey with the
Sudbury Wolves of the
Ontario Hockey League (OHL). On 4 April 2014, it was confirmed that Kahun had signed with
EHC Red Bull München to begin his professional career with the
2014–15 season in the
Deutsche Eishockey Liga (DEL). In 2016, he won the DEL title with München. On 21 May 2018, Kahun as an undrafted free agent signed a two-year entry-level contract with the
Chicago Blackhawks of the
National Hockey League (NHL). Kahun made his NHL debut on 3 October and recorded his first two NHL points, both assists, in a 7–6 overtime loss against the
Toronto Maple Leafs on 8 October. He recorded his first career NHL goal on 11 October in a 4–3 overtime loss to the
Minnesota Wild. He appeared in all 82 games with the Blackhawks in the
2018–19 season, providing scoring depth to Chicago, finishing tied 7th among all rookies in NHL scoring with 13 goals and 37 points. On 15 June 2019, Kahun was traded by the Blackhawks to the
Pittsburgh Penguins along with a 2019 fifth-round pick for
Olli Maatta. Kahun scored his first goal with the Penguins on 26 October 2019 against the
Dallas Stars. Kahun was injured on 19 January 2020, suffering a concussion during a game against the
Boston Bruins. He returned to the line-up on 18 February after missing 8 games. In 50 games with the Pittsburgh Penguins, Kahun added 10 goals and 17 assists. On 24 February 2020, he was traded to the
Buffalo Sabres in exchange for
Conor Sheary and
Evan Rodrigues. Kahun made six appearances with the Sabres, collecting two goals and 4 points before the regular season was initially paused then concluded due to the
COVID-19 pandemic. Due to salary cap considerations, Kahun was not tendered a qualifying offer by the Sabres during the offseason, making him a
free agent. On 1 November 2020, Kahun was signed to a one-year, $975,000 contract by the
Edmonton Oilers, joining countryman and league MVP
Leon Draisaitl. With the
2020–21 NHL season set to be delayed due to the pandemic, Kahun was loaned by the Oilers to return to his German club, EHC München of the DEL, on 6 November 2020, until the commencement of Edmonton's training camp. On 6 September 2021, Kahun joined
SC Bern of the
National League (NL) as a free agent on a three-year deal through the end of the 2023/24 season. On 22 August 2022,
SC Bern and Kahun agreed to another three year extension that runs through the end of the 2026/27 season. On February 18, 2025, it was announced that Kahun would play for the
Lausanne HC with immediate effect. His contract with the Lausanne HC lasts until the end of the 2026/27 season. ==International play==