Is a second ablution valid with the intention of removing impurity if the first ablution was valid and was renewed by mistake with the intention of removing impurity instead of the intention of renewing ablution?

There is no problem with the first ablution remaining valid, and not renewing ablution does not cause any harm. Even if it is renewed with the intention of removing minor ritual impurity, and the person was not in a state of impurity, that does not cause any harm.

As for the situation that involves a difference of opinion, it is when a person is in a state of minor ritual impurity and forgets his impurity, so he performs ablution with the intention of renewing ablution. This ablution is sufficient, according to the Hanbalis in the relied-upon narration, for removing minor ritual impurity and permitting prayer. As for the other narration, it is not sufficient, and it is the supported view among the Shafi'is.

Ibn Qudamah mentioned in Al-Mughni the two narrations' perspectives: 1. His purification is valid because it is a legal purification, and due to the report, and by analogy with the intention of removing minor ritual impurity. 2. His purification is not valid because he did not intend to remove minor ritual impurity, thus resembling the intention of cooling down.