All digital pianos will have some kind of mechanical extraneous sound, but the clicking from your digital piano is exceptionally loud. (Either that, or you recorded your example with the volume very low.)
One way to get around this extra noise while recording is to record the line out signal from the piano. You should be doing this anyway, unless you particularly like the sound through a certain keyboard amp. If your piano does not have line outputs, but does have a headphone output, you can use a splitter cable to plug both headphones and a digital recorder or PC to your piano. Specifically, you will need a "3.5mm headphone splitter" or "3.5mm TRS Y splitter, 1 male to 2 female".
For practice, you can use headphones to isolate you from the clicking sounds coming from the piano, if they bother you. If you're playing with other people, headphones may not be an option, but you're probably playing loudly enough (with a keyboard amp) that the clicking is not an issue.
As far as physically solving the problem at the source, solutions available will depend on the construction of your keyboard. Many keyboards will have a strip of felt that the keys touch when depressed all the way, which could be replaced, either with fresh replacement felt, or maybe some slightly thicker felt. I believe this is the replacement for the model mentioned in the question. I am not endorsing this seller. For weighted models, there may also be cushion strips for the hammers. Example Example If your keyboard has those cheap plastic-shell type keys, where the underside is accessible, then maybe you could try sticking some type of adhesive-backed gel or rubber pads in there to damp the keys.