What is the ruling regarding prayers that were performed while there was an impurity, which was not known about until after the prayer, and it was believed that the prayer was valid?

If someone prayed while unaware of the presence of impurity (najāsa), and then realized it after the prayer, their prayer is valid according to the soundest opinion among scholars. This is the madhhab of Malik and one of the two narrations from Ahmad, and re-performing the prayer is not obligatory upon them. Evidence for this is found in the Prophet's (peace be upon him) prayer while wearing his sandals; after Jibrīl informed him that there was dirt on them, he removed them but did not restart his prayer. This is also supported by Allah's saying: "Our Lord, do not take us to task if we forget or make a mistake" (Quran 2:286). This also applies to someone who prayed having forgotten about an impurity. There is no need to re-perform previous prayers that were offered in this state. Vaginal discharges are pure (ṭāhir) but invalidate ablution (wuḍū’).