Having played in their youth side for four years, Busk was promoted to the first team of F.C. Copenhagen during the 2012–13 season. He gained his first Superliga game on 16 May 2013, an away match against Randers FC.[2] In the Autumn of 2014, Busk was sent out on loan to fellow Superliga club AC Horsens, and the following year had a spell with Sandefjord.
On 14 May 2024, newly promoted Sønderjyske announced the signing of Busk from 1 July 2024. He signed a two-year contract.[4]
Busk made his debut for Sønderjyske on 11 August, coming on in the first half for the injured starter Nicolai Flø during a 2–0 league loss to his former club, Copenhagen. This was Busk's first competitive appearance in over six years, with his last game being on 7 April 2018 for Union Berlin.[5]