What is the ruling on touching phlegm and the large amount of impure water that exits the nose, and the effect of that on touching another person’s hand and on prayer, knowing that the nose is washed with water after wiping it with a tissue?

Mucus and discharge from the nose are pure, not impure. They are considered repellent substances. Their presence does not affect the prayer, and there is no harm in shaking hands with others if there is a trace of them on the hand. Simple sniffing and blowing the nose are sufficient without overdoing it.