You can do this via SSH. Login in via SSH and delete /accounts/1000/clipboard/text.plain. You can SSH via "Playbook Connect" or "Dingleberry" (this works without rooting the PB) and once SSH is running, you can connect using WinSCP if you want a GUI for browsing files. (You could also do it via the SSH terminal offered by both Dingleberry and Playbook Connect).
You can also edit the file to change the contents of the clipboard.
(You can also view the file by typing file:///accounts/1000/clipboard/text.plain into the browser, but you can't delete or edit it from there.)
EDIT: If you want more info on how to connect via SSH, I outlined the steps in a bit more detail here: Words with friends being creepy
EDIT: Another, easier method:
Get Dingleberry or Playbook Connect. Run either and open an SSH terminal, and type the following:
(sadly you only seem to be able to run scripts in rooted, otherwise you could easily run a little script to do this. Even so, it's not that difficult EDIT: Turns out you can run scripts without root, but it's a bit less convenient - easier to just type this one command)