You might think that when you hit the 'delete local content' button on steam it means that your are deleting all the local content for that game, right. Wrong! There are still residual files that are not only useless if the game is permanetley uninstalled but will also mean that things like addons are still pressent after a re-install.
This is how you can fix that:

Step 1: First we need to find our steamapps folder. Do this by oppening your Local Disk (ussually called C:) Next locate 'Program Files (x86)' (If you have no (x86)after the Prgram Files just click Program Files. This just means you are running a 32bit OS as opposed to 64bit.)
Step 2: Now open 'steamapps'
Step 3: Next, open the common folder in 'steamapps'
Step 4: Locate your GMod folder in 'Common'
Step 4: Simply delete the 'GarrysMod' folder. Don't worry as all your purchases are saved to steam, so you can just go back to steam and download again.

The full patth should look something like this:

This is the easiest step, just click library at the top right of your steam client. Find GMod and click the 'install' button.
Steam will now begin installing all the files you deleted, except this will be complete vanilla GMod, meaning, addons or any other random files that accumulate over time are no longer present.

Step 1: Exit steam and run it as administrator. Do this by right clicking the steam icon and click 'Run as Administrator'

Step 2: In your steam library, right click the game you need to validate and in the context menu open the preferences window.

Step 3: Navigate to the 'Local Files' tab at the top of the preferences window.

Step 4: Click the 'Validate integrity of game cache' button. A window will open. Note that this can take a long time if you have a low-end PC. The progress bar will often get stuck on 100% Don't worry, the software could still be computing at this time.

Step 5: If there were no discrepancies in the files, the window will close. If there were it should
display something in a window that you can show to whomever has asked you to perform this precedure, or you can post a topic on the steam forums to find help.

TIP: During this time you will be unable to use your steam client, seen as it can often take some time to fully validate the cache of a big game, you might want to find something else to do on your PC.