What is the ruling on the prayer of one who sprinkled water on his garment when madhy (pre-seminal fluid) was discharged, believing it obligatory to sprinkle water on it, then discovered that sprinkling is to be done on the private parts?

The method of purifying from madhiy (pre-seminal fluid) was mentioned in Fatwa 59304. The opinion that merely sprinkling water on it suffices is held by some scholars. If you purify your garment but do not perform istinja' (washing the private parts) from madhiy, then you are mistaken, and istinja' from it is obligatory upon you. As for the past prayers you performed without purification, there is a difference of opinion regarding their validity. You may follow the opinion of those who do not deem qada' (making up prayers) necessary in this situation.