[14] Following his provincial debut, Burke was selected by the Crusaders to play in the 2020 Super Rugby season,[15] operating as the third-choice fly-half behind Richie Mo'unga and Brett Cameron.
[7] He made his Super Rugby debut on 14 March 2020, coming on as a substitute against Japanese franchise Sunwolves, and scored nine points, including a try.
[23] Burke became a regular starter for the Crusaders in the 2023 Super Rugby season, although he played primarily at full-back, with Mo'unga continuing as the starting outside half.
[28] In 2024, Burke departed the Crusaders and moved to England to join Saracens on a long-term contract,[29] replacing outgoing fly-half and captain Owen Farrell.
[30] Following his first appearances for the club in two pre-season fixtures,[31] he made his competitive debut as a starter in the opening round of the 2024–25 Premiership,[32] kicking 10 points at goal during a win against Gloucester on 21 September 2024.
[33] This was followed by a European Champions Cup debut on 7 December 2024, when Burke started and scored seven points in a pool stage victory against the Bulls.
[41] Although he represented New Zealand at junior level, Burke also qualifies via ancestry, under the World Rugby selection eligibility criteria, to play for England or Scotland,[42] through his Dover-born mother and Glasgow-born grandfather, respectively.
[43] Burke ultimately chose to represent Scotland and, on 15 January 2025, he received his first call-up to the Scottish senior squad, ahead of the 2025 Six Nations Championship.