[4] After progressing the club's academy for the past three years, Kavanagh made his first team debut for Fulham in a 1–0 home loss on 20 August 2014 against Wolverhampton Wanderers at Craven Cottage.
[11] Though playing in midfield under the management of Kit Symons, Kavanagh made nineteen appearances and scoring once for the 2014–15 season.
[citation needed] In the 2015–16 season, Kavanagh struggled to be in the first team, having spent most of the time being on the substitute bench and was limited even further following the arrivals of James Husband and Luke Garbutt.
[12] On 21 November 2015, Kavanagh made his first appearance of the season, playing 90 minutes, in a 1–1 draw against MK Dons.
[23] After appearing once for Fulham in the 2017–18 season,[24] Kavanagh returned to Republic of Ireland, where he joined Shamrock Rovers for an undisclosed fee.
[31] After serving a one match ban, he returned to the first team from suspension, playing 90 minutes, in a 1–0 win over Sligo Rovers on 30 March 2018.
[36] However, in a 3–1 loss against Lithuania U21 on 13 November 2015, Kavanagh was sent-off in the closing stage of the game for a foul on Deimantas Petravičius.