Praise be to Allah.
If the imam makes a mistake in his recitation – apart from al-Faatihah – that does not alter the meaning, then you do not have to correct his mistake. This is something that no imam is entirely free from. But it is better to point it out to him after the prayer, because this will help him to remember it.
But if the imam refuses to correct his mistake even though he knows it is a mistake, then one to the following two scenarios applies:
1 – If the mistake alters the meaning, and he knows that it is a mistake but he does not correct his mistake after it has been pointed out to him, then you must repeat the prayer.
But how can you be sure that he realized it is a mistake and was persisting in it?
But if it is a mistake that does not alter the meaning, then you do not have to repeat the prayer, but you must advise the imam in a kind and gentle manner, and explain that it is not something shameful if the imam forgets something, for the best of Allaah’s creation, Muhammad (peace and blessings of Allaah be upon him), also forgot, and the Sahaabah who were behind him reminded him, as is mentioned in the hadeeth.
Shaykh Sa’d al-Humayd
With regard to correcting the Jumu’ah khutbah, this is one of the cases in which speaking during the khutbah is permitted, such as if the khateeb makes a mistake in reciting a verse that changes the meaning, such as omitting a phrase from the aayah and so on. In this case the one who corrects him is not regarded as coming under the prohibition on speaking during the khutbah. See al-Sharh al-Mumti’, 5/140.