Are the abundant white discharges—resulting from infections—after the cessation of post-natal bleeding on the eleventh day considered purity (ṭuhūr), and is ghusl and prayer obligatory, and is it necessary to make up missed prayers, or should one wait for the forty days to end?

There is no minimum limit for lochia (post-natal bleeding). Whenever a woman sees purity by the cessation of blood, yellowness (ṣufra), and brownish discharge (kudra), she must perform ghusl (ritual bath) and pray, and she will have all the rulings of pure women. White discharges are not considered lochia. What is considered is the cessation of blood, yellowness, and brownish discharge. She must make up the prayers she missed after becoming pure, thinking she was still in lochia, because they are a debt upon her.