I was having the same error tonight, and went a bit scorched earth on it but was able to get this working again.
Since I could not repair, repair online, or uninstall due to this error, I deleted everything in C:\Program Files\Microsoft Office, as well as C:\Program Files (x86)\Microsoft Office, C:\Program Files\Common Files\Microsoft Shared\OfficeSoftwareProtectionPlatform,
C:\ProgramData\Microsoft\ClickToRun, and also C:\ProgramData\regid.1991-06.com.microsoft\
Then also deleted registry keys for HKCU\Software\Microsoft\Office and HKLM\Software\Microsoft\Office and HKLM\Software\Microsoft\OfficeSoftwareProtectionPlatform
With all of that gone, it let me remove the control panel \ Programs entry with an uninstall (nothing really uninstalled, just said could not find and then removed).
I was then able to download and run a fresh Click-To-Run installer, and it installed fine but didn't ask me for a license key and also did not activate. Had to call microsoft but worked with the automated system and activated with a verification code.
All set. I'm sure less could be done, but this process has fixed up every dumb Click-To-Run error I've encountered by completely removing click to run from the computer.
Also, the site in question happened to use a Sonicwall TZ 210. This doesn't necessarilly have anything to do with Sonicwalls, but could have something to do with a Proxy Server if someone was doing web based content filtering on a Sonicwall (or any firewall
for that matter). Worth investigating as BGGF mentioned above, but not an end-all-be-all.