What is the nature of these continuous secretions, and are they from the Wadi, the Mathy (pre-seminal fluid), or the Maniy (semen)? What is their effect on prayer, and do they necessitate Ghusl (ritual bath)?

It appears that the secretions mentioned are the usual vaginal discharge. Ghusl (definition) (ritual purification bath) is only obligatory if one is certain of the emission of semen, which necessitates ghusl.

To understand the types of secretions and their rulings, the description of female semen (maniyy) and pre-ejaculatory fluid (madhy), the difference between them, how to deal with doubt regarding discharge, and the issue of continuous pre-ejaculatory fluid or vaginal discharge, one can refer to related fatwas.