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Rodri's injury is a big blow for Manchester City because he is an important player who has been in the main squad for years, there is always a disadvantage when both clubs play fiercely, especially Arsenal is Man City's enemy so they feel happy when Man City's main squad is injured I think this will be Arsenal's chance to take over the top position if Man City becomes unstable when Rodri is absent later.
But if Ilkay Gundogan can replace Rodri's role well, then Man City's performance will be fine and Pep Guardiola doesn't need to worry about Rodri's absence and hopefully his injury will not be prolonged.
Of course because Rodri is player who can always play very well as defensive midfielder or help every attack effort, last season he performed very well and contributed greatly to Manchester City achievements so that the injury he suffered would be very detrimental to the team.
It is reported that Rodri injury is quite serious and he is certain to be absent for quite long time in the future, but on the other hand you are right that Rodri injury problem will make it easier for several other teams such as Arsenal and Liverpool to try to reach the top position because they can take advantage of all this to be superior to Manchester City even though Arsenal is also in player crisis.
If only rely on Ilkay Gundogan, it will never be able to help and restore Manchester City performance to be perfect because he has his own place and is also one of the main players, there is still Mateo Kovacic as replacement but I not sure he can really replace Rodri and make the same contribution.
There will be possibility for Manchester City to bring in players to fill the void left by Rodri because the healing process after ACL surgery can take months to 1 year.