If you’re going there, leaving from the south exit is the fastest.
You can see a UFO at the southern entrance of this mall right now!
I was hit on at the south exit of Shinjuku station.
The South mouth isn't actually a mouth you find in the south pole or anything, it's an entrance or exit. This is kind of like how 入り口 means entrance (the 口 represents a doorway). You'll see this word a lot in places like train stations or airports. It means both South exit and South entrance.
The reading for these kanji are both kun'yomi. Probably because it's two words put together (South + Exit) rather than a single word. Also, body parts tend to change words into kun'yomi readings as well. So, you've seen vocab words that give you the kun'yomi reading of both 南 and 口, right? Now you just need to combine the two vocab readings into one to be able to read this word. Just watch out for the rendaku that changes くち to ぐち.