Liverpool's success in beating Manchester United with a score of 7-0 seems difficult to repeat, because at this moment Liverpool has a schedule with a difficult opponent. This at least forced Klopp to make a maximum rotation, with the rotation being carried out it would certainly make Liverpool's performance not as good as when Liverpool beat Manchester United.
Also, at this moment it looks like Manchester City will certainly try to beat Liverpool because in their first match or meeting this season Manchester City lost. Therefore, because in this match Manchester City will play at home, then of course Manchester City must be able to beat Liverpool as a way to take revenge as well.
Liverpool's instability makes it difficult to analyze the possible outcome of the match as a whole. It seems obvious that Manchester City is really stronger, and they need a win more than ever, unlike Liverpool, who already have nothing to lose. But on the other hand Liverpool is still a top club, so it's hard to say that they will lose this match easily
And like you said, I think Manchester city need the win more then Liverpool, reason being that, if they can't climb to the number one position, they need to atleast, maintain their second position else, they drop to the third place, which would be a big disappointment in my opinion.
I doubt they can go down to the third line. Manchester City have already made good progress and are racing behind Arsenal, I think at the top of the table there is only this race, the rest are not so important, including for many fans. Liverpool have a slightly different situation, at the moment they can 100% fight for a place in the Europa League, but also have a small chance to get to the Champions League, if they try very hard