I've just read that you shouldn't use bleach because the water content of it goes into the surface and the deeper mould spores feed on that. I've discovered that the small dots on the inside of my (unpainted) front door are black mould - not mud splats as I previously thought 
I'm waiting for warm weather to dry the door thoroughly before painting. I'm currently living without central heating, hence the mould, though it's very small so far.