Translation:He doesn't bathe because of the water shortage.
"He doesn't wash himself because of a shortage of water." Would "because of a shortage of water" be equivalent to "because of the water shortage"?
I think "a" instead of "the" can be ok, depending on the context.
"The" implies that it is a specific water shortage that both speaker and listener is aware of.
"A" would be appropriate only if the listener is not aware of the water shortage.
Hope this makes sense.