Is it permissible to perform tayammum for obligatory and supererogatory prayers when there is continuous discharge due to vaginal fungal infections, making it difficult to maintain wudu’? And is it permissible to touch the Mushaf with a barrier in this situation?

If the discharge is continuous, its ruling is the same as that of urinary incontinence. You should perform ablution for prayer after its time has begun, and pray with that ablution as many obligatory and supererogatory prayers as you wish until the time for the prayer expires. It is not permissible to perform tayammum (definition) (dry ablution) as long as you are able to perform wudu' (ablution with water). These discharges are pure and do not necessitate istinja (definition)' (washing the private parts).