I have several of this same model and it seems dust gets in within a year or less no matter how carefully I try to keep them in a pouch when using them. For me, the black specks are dust on the CCD. Here are two links I found useful in opening the back cover, getting to the CCD and cleaning it. A problem I’m not able to address is dust inside the lens assembly. Apparently the only solution I’ve seen is to replace the lens assembly. iFixit instruction for opening the camera Sony Cyber-shot DSC-HX20V Outer Case Replacement and the YouTube video for a different model but the last few minutes are applicable for cleaning the CCD https://www.youtube.com/watch?v=774HX_-iUGI
Step 5 - perhaps a little more accurate step would be to lift up the bottom of the panel and then slide the panel downward. (reinstall in reverse direction).
I am having trouble orienting the control wheel - I tried to seat the middle button so it clicks and then set down the wheel on top but that does not seem to work after putting the back cover on. I’ve reopened and tried many times. Any suggestions on how to orient or reinstall the control wheel.