Must the prayers performed on a towel containing impurity, of which one was unaware until after performing a number of فرض (prescribed prayers), be repeated?

If the towel is impure and you pray on the pure part of it, the prayer is valid and there is no problem, as mentioned by An-Nawawi and Ibn Qudamah. And if the worshipper directly touches the impure part of the towel during prayer, the prayer is valid according to many scholars, and this is the relied-upon opinion.