Removing the cover art is easier than correcting it, but I'll cover a few options.
Removing all album art from Google Play: From the songs page, select every track. Right-click within the selection and select 'Edit Info' from the pop-up menu. On the cover art icon, click the X to delete the cover art from all your songs.
Removing just the bad album art: Create a playlist for albums with bad cover art. Go to the Albums page and add the ones with bad art to the new playlist. Select all the songs in that playlist and delete the cover art as above.
Actually solve the problem: First, you need to edit the ID3 tags on your MP3s. (I'd be surprised if these were not the source of your album art troubles. Edit: Looks like I was completely wrong about that. Seems Google Play has some album art problems and they know it. Mea culpa.)
If you have Windows, download Album Art Downloader from SourceForge at http://sourceforge.net/projects/album-art/. HowToGeek has a nice step by step process for correcting your MP3 library's album art.
If you are on OS X or Linux, it is a bit trickier because there is not one standout bit of software that I've used to fetch and embed album art. iTunes, for example, does not automatically embed album art in your ID3 tags and the process for doing so depends on what version of iTunes you are using. Blegh. However, the song metadata at MusicBrainz is pretty good and they have a cross-platform-ish ID3 editor called MusicBrainz Picard that, with a plug-in called Cover Art Downloader, will grab covers for all your albums and embed them as needed.
You still need to delete cover art in Google Play to get the new art to update. In fact, you may need to delete entire tracks or albums and re-upload them with the new ID3 tags. In researching this I found the latter method to be effective and sufficiently quick, but your mileage may vary.