Remember, this is a small sensor camera, so dust is magnified. With the note that it is invisible at wide aperture and visible stopped down, the diagnosis is near-certain: there's a speck of dust on the low-pass filter in front of the sensor. (There's a small chance that it's a stain from water as Boby says, as well, but the sample looks more like dust to me.)
With a compact camera, sending it in for cleaning is usually the only option. This'll probably cost $100 — and they'll probably do it under warranty for you at least once. If you don't want to pay that, you can:
- resign yourself to only shooting with wide apertures in situations where the dust isn't visible
- clone it out in post-processing every time
- use this as an excuse to buy an interchangeable lens camera with a dust-removal system (and where you can clean the sensor yourself if need be)