"File" is damaged and can't be opened. You should move it to the trash.

Start by trying the following:

Repair the Drive for El Capitan or Later

1. Restart the computer. Immediately, at or before the chime, hold down the Command and R keys until the Apple logo and progress bar appear. Wait until the Utility Menu appears.
2. Select Disk Utility and press the Continue button.
3. Then select the indented (usually, Macintosh HD) volume entry from the side list. 
4. Click on the First Aid button in the toolbar and wait for the Done button to activate. Click on it, then quit Disk Utility.
5. Select Restart from the Apple menu.

Try opening the files. If the problem persists, then try recovering them from your backups. If the problem still persists, then see All downloaded files cannot be opened as they are 'damaged ... for a possible solution. Lastly, I suggest the following:

Check if the com.apple.quarantine flag is set, and if so, remove it. Open Terminal in the Utilities folder and enter the following commands at the Terminal's prompt - simulated by the arrow. Press RETURN after each command. If there is no response from the first command, then the flag is not set and you are done.

➜  xattr foo.pdf
com.apple.quarantine (this is the response from the first command.)
➜  xattr -d com.apple.quarantine foo.pdf
➜  xattr foo.pdf

If you still have the problem, then try modifying Gatekeeper in Security & Privacy preferences so you can open Anywhere:

Again, in Terminal paste the following at the prompt: sudo spctl --master-disable and press RETURN. Open Security & Privacy, click on the General button. You should now see at the bottom a radio button labeled Anywhere. Enable it by clicking on the radio button.