What is meant by wearing socks while in a state of purity (ṭahārah), and is it permissible to wipe over them if they were worn after performing ablution (wuḍūʾ), then the ablution was invalidated, and it was not renewed before wearing them?

For the validity of wiping over the socks (masah alal-khuffayn), it is a condition that they be worn after completing a state of purity (taharah). The intention here is wearing them after washing all the limbs required for ablution (wudu'), as clarified by Sheikh Ibn Uthaymeen. So, if one were to wear the sock on the right foot before washing the left foot, wiping would not be permissible, because that is before the completion of purity. If one were to wear both socks after an impurity (hadath) and before repeating ablution, then they were worn in a state of impurity, and it is not permissible to wipe over them. In this case, one must remove them, perform ablution, and then wear them again if one wishes to wipe over them in the future.