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