A boyhood Linfield supporter, Rhys is a striker who hails from Newtownards. Annett started out playing youth football for Ridgeway Rovers before moving to Dundela where he came through the youth setup before making his debut. Incidentally, Annett scored against Linfield on his Dundela debut when he was 16 in a 6-1 Co. Antrim Shield defeat at Windsor Park. In total, the teenager scored 12 goals in 43 appearances for the Duns.

He arrived at Windsor Park in January 2023 as part of a double swoop along with Ryan McKay. Annett made his Linfield debut on March 24, 2023, although it was the only appearance he made for the first team in his first season with the Blues. Despite that, having initially signed as an amateur, he then signed a pro two-year contract extension in June 2023 and made more of an impact in his second season, scoring five goals in 26 appearances – one of which came in the NIFL League Cup final in a victory over Portadown.

Rhys signed a two-year contract extension with The Blues which will see him stay at the club until the summer of 2027.