Burgess began his career with the Shakers as a youngster and signed his first professional contract in 2014. His senior debut in football came on the final day of the 2013/14 season where he started in a 0-0 draw with Morecambe in League Two.
In September 2015 he was loaned out to Stalybridge Celtic in the National League North where he made three appearances in a month long spell.
Upon his return to Gigg Lane he once again faced Morecambe, this time in the Football League Trophy and made a further four appearances for Bury in League Two.
In March 2016 he was loaned out to Norwegian side Bergsoy, where he made 14 appearances and grabbed his first senior goals, four in total during his time abroad.
His first goal on British soil came in December 2016 against Bristol Rovers at the Memorial Ground, Bury eventually lost 4-2. Two weeks later he scored his second during a great run in the team to earn a point in a 1-1 draw away to Bradford City.
After 17 appearances during the 2016/17 season he was loaned out to Macclesfield Town in the summer. His Silkmen debut came at his future club Wrexham, where he was part of a 1-0 victory for Macclesfield at the Racecourse.
He would make over 30 league appearances in a season that saw Burgess and Macc lift the National League title and earn promotion to the EFL.
In the summer of 2018 Burgess linked up with the Dragons signing on loan until January.