Does continuous vaginal discharge—whose exact time of emission is unknown—count as an excuse that permits performing ablution for every prayer and praying without confirming that anything has been emitted, or is it necessary to repeat the prayer due to the possibility that nothing was emitted during it?

A woman who suffers from continuous discharge or irregular periods should perform ablution for each prayer after its time has begun. With that ablution, she can pray the obligatory and supererogatory prayers until the time for that prayer exits. There is no need for excessive investigation or scrutiny. The discharge is pure and does not necessitate istinja (washing the private parts), but it does nullify ablution.