Don't cut a wide one
Picked up a story yesterday in which the writer had written swathe for swath. This is a muddy patch in the language, and the Old Prescriptivist is going to lay down his coat so that you don’t have to get your shoes dirty.
Use swath as a noun, meaning a strip of space. Its origin in Middle English indicated the amount of space covered by a single stroke of a scythe. The metaphorical expression cut a wide swath means to make a big impression or to indulge in some ostentatious display. Pronounce it “swahth.”
Swathe is a noun for a bandage or wrapping. Put it out of your mind that it is an alternative spelling for swath; that will not help you. It is more frequently used as a verb, pronounced “swaythe,” meaning to wrap a strip of bandage around something, or, metaphorically, to surround or enclose. It’s akin to swaddle.
Keep them separate
This is the sort of point on which the Associated Press Stylebook could be useful, if it had an entry on this point. (Perhaps you could suggest it to them; I lack confidence that they listen to me.)