What is the ruling on a white, viscous fluid exiting after urination by some time, and what are the implications of that for acts of worship that require ritual purity?

The preponderant opinion is that what comes out after urination is "al-Wadi," which is a thick, white fluid that usually exits after urination. The ruling regarding its purification and ablution (Wudu') after it is the same as that for urine.

Whoever experiences the exit of al-Wadi, or any other nullifier of ablution, after urinating, must wait until they are certain that its discharge has stopped. Then, they should perform istinja' (washing the private parts), perform Wudu', and pray. Ablution is only nullified by the exit of something certain. If the exit is certain, then istinja' and Wudu' are obligatory thereafter, as long as it accompanies him for less than half the time, and it is obligatory to purify the place of impurity on his body and clothes.

If he prays before anything exits, his prayer is valid. However, if he is certain that something exited after Wudu', or before the prayer, or during it, then he must perform istinja', repeat Wudu', and repeat the prayer, according to the saying of the Prophet, peace and blessings be upon him: "Allah does not accept the prayer of anyone of you if he passes wind until he performs ablution."

One must be cautious of obsessive doubt (waswasah). If what you are speaking of is merely doubt or waswasah, then pay no attention to it.