In the mentioned situation, the bleeding that occurred between December 29th and January 2nd in the form of a yellowish discharge will be considered as postpartum bleeding (Nifas). The bleeding that occurred on January 10th will be considered as Istihazah, as it came before the minimum duration for purity (Taharah) of fifteen days. The ruling for Istihazah is that for each prayer, perform a new ablution (Wudu) and offer the prayer.
After completing the full forty days, if the bleeding stops within ten days, then from January 16th onward, the complete ten days will be counted as menstruation (Hayd), and it will be considered that the menstrual cycle has started anew. If the bleeding exceeds ten days, the blood that comes within the days of the usual menstrual cycle will be counted as menstruation, and the remaining days will be considered as Istihazah