I've been running v3.8.0 on 10.9.3 and just upgraded to 10.9.4 this morning and the Kobo Desktop App is running without issue.
You mention that you uninstalled the app. Did you just delete the app, or any associated files too?
Can you try the following:
1. Ensure the Kobo app is not running.
2. Delete the Kobo app from the Applications folder
3. Switch to Finder and go to the "Go" menu and select the "Go to Folder..." option, enter this in the dialog that appears
~/Library/Application Support/
4. Locate the "Kobo" folder and delete it
5. Switch to Finder and go to the "Go" menu and select the "Go to Folder..." option, enter this in the dialog that appears
~/Library/Preferences/
6. Locate the file named "com.kobo.Kobo Desktop Edition.plist" (depending on how you have things set up, you may not see the .plist portion of the filename) and delete it.
7. Ensure the trash is emptied to fully remove these files
8. Download the latest Kobo Desktop app from here:
http://www.kobosetup.com
9. Mount the disk image and copy the Kobo.app file to the Applications folder
Double click on the Kobo.app in the Applications folder and see if it launches.