Some people will tell you, mould killing chemicals, but that's wrong.
The entire world, and the air above it, are full of mould spores.
Why isn't my wall mouldy? Because it isn't damp.
Why is your? Because it is.
You didn't mention which room(s) this is in.
If a kitchen or bathroom, where water functions are performed, you need an effective extractor, and you need to use it.
There is the possibility of a water leak if the room contains a tap, a drain, a pipe, a boiler or a radiator; or if any pipes or drains are in the floor, especially if it is concrete.
If the extension was very badly built, the floor may be damp, depending on its method of construction, and the roof may leak, or the gutters.
Does "one brick" mean four inches thick, or nine?
The one thing that I can be sure will not fix the ventilation or the leak, is a chemical injection into the walls.
post some photos and explain what part of the wall is mouldy.