Is it sufficient to pass the hand over the penis and cover it with water to wash away pre-seminal fluid (madhy), and has a specific method for this been authentically reported from the Prophet, peace and blessings be upon him?

Pre-seminal fluid (madhy) is impure and invalidates ablution. To purify oneself from it, it is sufficient to wash only the affected part of the male organ, not the entire organ, according to most scholars. There is no need to drench the entire organ with water while passing the hand over it.