What is the ruling on the prayer of one from whom drops of urine or madhy have exited, if he is able to wash his penis but not his clothes due to embarrassment, or if he is unable to wash either of them due to the unavailability of a private place?

As for a small amount of madhy (pre-seminal fluid), the preponderant opinion is that it is excused, especially for young people, based on the hadith of Ali concerning madhy: "And you take a handful of water and sprinkle your garment with it." As for drops of urine, the default is that a small amount of urine is not excused except in cases of extreme hardship, and the garment must be purified. Shyness is not an excuse for abandoning ritual washing, based on the Prophet's (peace be upon him) saying: "If one of you invalidates his ablution during his prayer, let him hold his nose and then depart." If purifying the garment is impossible without uncovering one's awrah (private parts) in front of someone before whom it is impermissible, then you are not allowed to uncover it, and the prayer must be performed on time, and your prayer is valid.