What is the ruling on beginning purification and prayer, then having intercourse after a slight discharge of blood mixed with secretions, especially with the use of an IUD and the change in the nature of the menstrual cycle?

The slight bleeding you observed is considered menstrual blood because it connected with what came after it, thereby exceeding the minimum duration of menstruation. You should have performed ghusl immediately upon seeing the purity. You must repent and make up the missed prayers. There is no expiation required from you or your husband if the intercourse occurred during a period of purity, but both of you must repent. You should strive to learn the rulings of menstruation, as yellowish or brownish discharge preceded by purity is considered purity, while if preceded by menstruation, it is considered menstruation. The maximum duration of menstruation is fifteen days. If the bleeding and purity exceed this period, then you are considered to have istihadah (non-menstrual vaginal bleeding), and your previous habit or the distinguishing characteristics of the blood should be relied upon. Otherwise, the common habit of women is six or seven days.