Are brownish discharges that are excreted all day due to exhaustion considered istihadah (prolonged non-menstrual vaginal bleeding), and what is the meaning of "the excretion of these discharges is added to the nearest time they are likely to have been excreted" given the presence of normal white discharges?

Continuous discharges are considered akin to سلس (incontinence), so the woman experiencing them should perform ablution for each prayer after its time has begun, and she may pray as much as she wishes with this ablution until the time for that prayer expires. White discharges are pure but invalidate ablution. As for yellowish and brownish discharges (صفرة وكدرة), they are impure and necessitate Istinja' (cleansing with water). If seen, they are not considered menstruation, so one should perform Istinja' from them and make ablution for each prayer. The rule of attributing an event to the nearest probable time applies to someone not afflicted with سلس (incontinence). So, if a person invalidates their ablution and does not know when, it is estimated to have occurred after the prayer, as that is the nearer of the two times.