What is the white powdery stuff that leaks out of a battery if I accidentally submerge it in water?
In the case of e.g. a lead acid battery the powder is probably lead/zinc/alimuinium sulphate. In an alkaline battery it might be potassium hydroxide.
With most batteries (e.g. batteries used in consumer products) it is usually a sign of failure (leakage) and the need to replace/dispose of it.
With e.g. a car lead acid battery I don't think this is always the case, sometimes it may just need some attention as small amount of corrosion on the terminals is often expected and can be dealt with.