How can one deal with continuous pre-seminal fluid (madhiy) discharge that may last for a full day or more, and is prayer obligatory in this situation, especially if its cause is masturbation?

O questioner, you are in grave danger by abandoning prayer. It seems this is due to masturbation and the continuous discharge of pre-seminal fluid (madhy), even though the emission of madhy does not permit abandoning prayer. If madhy is discharged continuously or most of the time, then you are considered to be in the same ruling as a person suffering from incontinence (sahib al-salas); you should perform ablution (wudu') when the time for prayer enters, and pray the obligatory prayer and its supererogatory prayers. What is discharged during the prayer will not harm you. What is obligatory upon you is to repent to Allah, return to prayer, be diligent in observing it, and make up for any prayers you have missed.