Home games give the home team an advantage of course. I am sure that a quick check for the statistics would show that Real Madrid is stronger when they play in the Bernabeu stadium than when they play away from home, but this isn't surprising at all.
But of course there are limits to the effect that the support can have. The missing players could become a problem, but luckily for Real Madrid, Manchester City is far from being the dominant force that they used to be in the past. Chances are still good for them to advance to the next round.
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As a host of a second leg match, the host Real Madrid will have advantage and different plans to do in this pair, but they have to avoid bad result in the first leg match in Manchester tomorrow. Any club that has bad result in a first leg match will have big troubles in the second leg, and if assumed Real Madrid have a big defeat in Manchester City, they will return to Madrid and host a second leg match with massive disadvantage and high risk to lose it to Manchester City again.
Man City itself is also not as strong as the previous season and there are the same weaknesses in the defense line so that both teams will try to cover up these weaknesses. Real Madrid can take advantage of this match to get a win but Man City will not let them easily get three points at home.
Manchester City are weaker than themselves in early of season before Rodri injury and previous seasons, but they are now not too weak to easily beaten by Real Madrid. They can collapse like in the match with Arsenal, but they can play strongly in a match with Chelsea, it's how volatile and unpredictable Manchester City now, and it's a tough math for Real Madrid too.