Praise be to Allah.
Continuous discharge comes under the heading of continual impurity; it is like istihaadah (non-menstrual vaginal bleeding), urinary incontinence or continual passing of wind. The one who is affected by this discharges has to do wudoo’ at the time of every prayer, then pray as much as she wants of naafil prayers, and she may touch the Mus-haf with that wudoo’, until the time for the next prayer begins.
With regard to reciting Qur’aan without touching the Mus-haf, wudoo’ is not required for that. The person who is impure in the sense of minor impurity may recite Qur’aan by heart, or recite from the Mus’haf if he holds it with a barrier such as gloves and the like. For the ruling on discharges please see question no. 50404.
What is meant by the beginning of the time is the time when the adhaan is given for the prayer, according to the timing in each land.
Islam has stipulated the timings for the beginning and end of each prayer. You will find details of this in the answer to question no. 9940.
If the time for ‘Asr, for example, starts at 3 p.m., then when the woman who is experiencing discharges does wudoo’ for Zuhr, she may touch the Mus-haf, circumambulate the Ka’bah and offer as many naafil prayers as she wants until 3 p.m., when the adhaan is given for ‘Asr prayer, so long as she does not break her wudoo’ in any other way apart from discharges during that time, such as by urinating and other things that are known to break wudoo’, because she is only excused for the discharge because it is continuous and it is hard to stop it.
And Allaah knows best.