Praise be to Allah.
This sticky white substance comes under the same ruling as other secretions, and is subject to further discussion.
If it is continuous, then it comes under the same ruling as urinary incontinence, so you have to do wudoo’ for each prayer after the time for it begins, and these discharges will not matter after that, even if that happens during prayer.
If it is not continuous, then there is a difference of opinion as to whether it invalidates wudoo’. The majority of scholars are of the view that it does invalidate wudoo’, but some of the scholars are of the view that it does not invalidate it, because there is no evidence to that effect. But to be on the safe side, one should follow the view of the majority.
This discharge is taahir (pure) according to the most correct view, so there is no need to wash them off one’s clothes. This is the view of Abu Haneefah and Ahmad, and it is one of the two views narrated from al-Shaafa’i, and was regarded as correct by al-Nawawi.
This view was also favoured by Shaykh Ibn ‘Uthaymeen. (May Allaah have mercy on them all.)
He said in al-Sharh al-Mumti’ (1/329): If these discharges come from the vagina, then they are taahir (pure), because they are not waste products of food or drink, so they are not urine. The basic principle is there is no najaasah (impurity) unless evidence is established to that effect. And if a man has intercourse with his wife, he does not have to wash his penis or his garment if anything gets onto them; if it were naajis (impure) then that would imply that maniy (semen) would become naajis, as he result of being contaminated with it. End quote.
See also al-Majmoo’ (1/406); al-Mughni (2/88).
These discharges do not affect the fast.
To sum up: you may fast and pray, and you should do wudoo’ when these discharges come, unless they are continuous and last all day, in which case you should do wudoo’ for each obligatory prayer when the time for it begins, and you can pray with this wudoo’ whatever naafil prayers you wish, until the time for the prayer for which you did wudoo’ ends.
And Allaah knows best.