City's main backer Etihad Airways also produced a short video of congratulations which featured players Claudio Bravo, Danilo and Ilkay Gundogan.
Kun will miss City's five remaining fixtures, but is expected to be ready to go at this summer's World Cup where he will represent Argentina.
Aguero made another brief cameo appearance during City's vain attempt at a Champions League comeback against Liverpool last week, but manager Pep Guardiola ruled the 29-year-old out of Saturday's 3-1 win over Tottenham, citing Young's challenge as the cause of the injury.
Sea of Thieves Content Updates Coming Soon
The Hungering Deep , the game's first content update, will be released in May along with a new AI threat and event. Some new mechanics will be added and unique rewards for completing the event are incoming.
And with five games remaining in the season and no other silverware to play for, it was chose to have an operation now.
The City boss said: "I don't know (when he will be fit)".
"This season has been incredible and the football we've played has been the best I've been involved with while at the club". "It's been a pleasure to play with this team and be part of everything we've achieved this season".
City are set to receive the Premier League trophy after their final home game of the season against Huddersfield on 6 May. "Hopefully (he will be back) for the last games, if we need him, and especially for the World Cup". "Let's enjoy the moment together with our fans and finish strongly".