Is it obligatory to perform ghusl again for a woman who saw a mucous-like discharge after four days of her menstrual period, and considered it menstruation, then a doctor informed her that it was istiḥāḍa, if she had already ascertained ritual purity and purified herself?

If the blood occurs during a period when it could be menstruation, and the total duration of the bleeding, the returning blood, and the purity in between does not exceed fifteen days, then it is menstruation, and you must perform ghusl after it ceases. However, if it exceeds fifteen days, then it is Istihada (non-menstrual vaginal bleeding).