This recipe for challah bread calls for an egg wash composed of 1 egg and 1 tsp of water (see the final item in the ingredient list).
What is the effect of this small amount of water? It seems like most other challah recipes use egg alone.
Egg white is very gloopy, unless you cut it with some water or milk it's hard to get a good, even layer. With challah this can mean you get a blob of egg white running down into a seam between braids which doesn't look good. A bit of water loosens it up and makes it much more spreadable.
If anything, for egg washes in general, it might be more common to add water (or milk or cream) than not.
Thinning the wash makes it easier to brush on an even, not-too-thick layer. It also tends to get you a more golden color, less dark brown.