Peterborough United Burrows joined the
Peterborough United academy at the age of six after attending a development centre. Burrows went on to feature in a pre-season friendly against
Ipswich Town in 2017 at just 15 years old. The midfielder signed his first professional contract in January 2019, making his debut for the club in August 2019 in an
EFL Cup match against
Oxford United. He went on to make his full league debut, aged just 17, in a 4–0 win against
MK Dons. During the
2023–24 season, at the age of 22, Burrows was named as Peterborough's club captain. On 2 November 2024, Burrows scored his first goal for
Sheffield United in a 2-0 away win over
Blackburn Rovers at
Ewood Park. ==Career statistics==