What is the ruling on delaying prayer when vaginal discharge occurs and one is unable to distinguish between what is pure and what nullifies ablution outside the home, and what is the solution to this problem, especially with the presence of obsessive thoughts?

Vaginal discharge (vaginal moisture) invalidates ablution (wudu') but is pure (tahir) and does not necessitate istinja' (washing the private parts) nor purifying clothes. There is no need to inspect for it. If you are certain it has exited, perform wudu' and pray, and do not delay the prayer. If its discharge is continuous without interruption, you may perform wudu' after the time for prayer has begun and pray with this wudu' as much as you wish until the time for prayer expires. It is permissible to combine two prayers due to hardship.