What is the ruling concerning the prayers that preceded the emission of madhy (pre-seminal fluid), knowing that the questioner did not perform ghusl (ritual bath) until some time after realizing the emission of madhy?

Ghusl (definition) is not required for the emission of madhy (pre-seminal fluid). Rather, it suffices to wash whatever part of the body or clothing the madhy has touched, and to perform wudu'. The method of washing it is by immersing it in water. There is no need to repeat past prayers that were performed with the impurity of madhy due to ignorance or forgetfulness.