How does a Muslim purify himself if madhy (prostatic fluid) is discharged merely by thinking about or being close to his wife, and its discharge continues for a long period that may overlap with prayer times?

If what you experience does not reach the level of سلس ( سلس البول - urinary incontinence/constant discharge), then you must wait for the discharge to cease before praying, even if you miss congregational prayer. For certain purity takes precedence over congregational prayer. You should also avoid anything that arouses desire (sexual arousal) to perform the prayer correctly. If you perform ablution (wudu') and pray, then doubt whether madhy (pre-seminal fluid) was discharged, do not pay attention to the doubt. If you are certain of its discharge but doubt the time of its occurrence, attribute it to the latest possible time. If you are certain of its discharge during prayer, then repeat your prayer.