You're sure it isn't in the (original) recording? With speakers I had to listen close up. The most prominent click seems to be the one about 1 minute in. I decided to go back to headphones to check this. I also reversed the phones as I think my left ear may be more sensitive. I agree that there are clicks, though the piano itself sounds clangy, and there are other noises going on in the recording - maybe pedalling. These are very obvious after the click at about 1 minute. I did wonder if there wasn't a click on the other channel too, and of course there are clicks at the end of the snippet. You seem to find them annoying, and I wouldn't disagree that they are predominantly on the right channel during the music.
Do you only get the clicks during music? I suppose once you know they're there you do listen for them - and, yes, they do start to get annoying.

