Bodø/Glimt Born in
Bodø,
Nordland, spending his childhood years in FK Vinkelen, and progressing through the youth academy of
Bodø/Glimt, Hauge signed his first professional contract with the club on 12 April 2016. The following day he made his senior debut in a 6–0
cup win against
IF Fløya as a 64th-minute
substitute for
Fitim Azemi. He immediately scored a
hat-trick. Hauge made his league debut on 23 April against
Strømsgodset, where he came on as a late substitute as Bodø/Glimt lost 2–0. In the next league games he featured regularly, and on 8 July, he scored his first top-flight goal in the 4–1 away win over
IK Start, becoming Bodø/Glimt's youngest ever goalscorer in the
Eliteserien. In the 2016 season, he made 20 league appearances in which he scored one goal, as Bodø/Glimt suffered relegation to the second-tier
Norwegian First Division. In the following season, Hauge established himself as an undisputed starter as Bodø/Glimt won the second-tier championship and made their immediate return to the top-flight Eliteserien. He made a significant contribution that season, scoring two goals and making 13 assists. At the start of the 2018 season, Hauge again lost his starting role, and as a result, was loaned to the second-tier club
Aalesunds FK on 15 August 2018, halfway through the season, after 11 league appearances in which he scored no goals. That season, he made 28 league appearances in which he scored seven goals and registered two assists. After ten match-days, the club surprisingly took a lead in the league table. At that point, Hauge had already scored six goals and assisted six. As a result, media reported that the Rossoneri were interested in signing him.
AC Milan On 1 October 2020, Hauge signed a five-year contract with
Serie A club A.C. Milan. Three days later, he made his debut in a 3–0 win over
Spezia. On 22 October, he scored his first goal for Milan in a 3–1 win at
Celtic in the group stage of the
Europa League. On 22 November, he scored his first Serie A goal in a 3–1 away win over
Napoli.
Eintracht Frankfurt On 10 August 2021,
Eintracht Frankfurt announced that Hauge would join the
Bundesliga club on a season-long loan with an option to make the deal permanent. He scored a goal in his Frankfurt debut, although in a 5–2 loss against
Borussia Dortmund, on 14 August. On 28 May 2022, Eintracht Frankfurt announced that they had activated the purchase option for Hauge from
Serie A club
AC Milan following his one-year loan. Hauge signed his contract until 2026.
Loan to Gent On 16 August 2022,
Gent announced that Hauge would join the
Belgian First Division A club on a season-long loan.
Return to Bodø/Glimt On 31 January 2024, Hauge returned to
Bodø/Glimt on loan until the end of 2024, with an option to buy. On 16 January 2025, he joined the club on a permanent deal by signing a contract until 2028. He scored his first
UEFA Champions League goals by netting a brace in a 2–2 draw with
Tottenham Hotspur on 30 September 2025. During the
2025–26 Champions League knockout play-offs, he scored in both legs and provided an assist in 5–2 victory on aggregate over
Inter Milan. ==International career==