If it is the sensor that's dirty, the first thing to do is try to blow off the dust with a strong blower. If it doesn't work, you'll need to clean the sensor with a swab and sensor cleaning solution, if you plan to do it yourself. You can get the swab and the Eclipse cleaning fluid from CP. But you'll need to be extra careful not to scratch your sensor when cleaning, so make sure the swab is absolutely clean and without any sand particles on it. Also, you will need to open up the shutter blades to perform the cleaning/blowing. Make sure that you're either using a power adaptor, or at least the battery is fully charged, as if the blades close while cleaning, the shutter will be damaged. But these are just some precautions, don't worry too much, just be careful. If not, just send your camera in to Sony Service Center and let them do it for you.
I believe it's dust on your cam's sensor. Not to worry, it's very common. This can be solved by using a blower most of the time.
Be careful not to put the blower too close to the sensor as the tip of the blower may scratch the sensor and replacing that would be expensive. As recommended, the Giottos rocket blower is good enough. Try getting a quote from Cathay Photo or MS Color.
As Sony cameras have the feature to shake the dust off everytime you on/off your cam, you can try switching the cam on/off a couple of times to see if it's better and remember to switch off your cam before changing lenses and always face the cam down when changing lenses. Also, avoid changing lenses in dusty places (if possible). These will help to reduce dust from getting into your cam.
As to your question on putting on a bigger filter on your lens, yes, it's possible. Just need to get a step up ring will do.
Minolta. Konica Minolta. Sony