Can't speak to the cloudy spot, especially without knowing what it is. Is it haze (perhaps in a sealer)? Something that can be "scraped" off? A mineral deposit from hard water?

However, here's how I clean my granite when I really want it to shine:
1 - wipe down the whole thing with a slightly-soapy sponge/dishcloth, followed by a clean barely-damp sponge/dishcloth to get up any soap left. (on an average day, this is where I stop)
2 - Use a clean dry towel to wipe up all the water left by the sponge, if necessary
3 - Lightly spritz a mixture of rubbing alcohol and water over the whole surface (I keep a spray bottle under the sink, premixed. About 1:4 alcohol:water)
4 - Wipe with a clean dry towel (may be able to use another side of the towel used in step 2). The benefit of the alcohol mixture is that it dries faster than plain water, and is less likely to leave streaks/spots.