As part of Apple's efforts to hamper the proliferation of dubious applications within its ecosystem, and more specifically on Mac devices, the tech giant introduced several security features. Most of the work is done in the background and whenever a suspicious activity is found, the macOS triggers an alert prompt that is shown to the user. The text of the warning usually states that '[App name] will damage your computer.'
In the vast majority of cases, the message signals that a PUP (Potentially Unwanted Program) has managed to sneak on the Mac and it should be removed as soon as possible. However, under certain conditions, the macOS may flag even legitimate applications and show this message incorrectly This is exactly what has happened to users who have encountered the 'Cisco AnyConnect Secure Mobility Client will damage your computer' warning.
What has most likely happened is that there is an issue with the current application signature - it may be outdated or it could have been flagged by Apple for some other reasons. The solution is quite simple as all that users need to do is to upgrade to a newer version of the application that has been signed properly. Afterward, the warning will disappear.
If you do not have the Cisco AnyConnect Secure Mobility Client installed while seeing this warning prompt, then things could be a bit more serious. You could have a PUP or malware on your Mac that is impersonating the legitimate application. Initiate a scan of the Mac with a professional anti-malware solution as soon as possible, and see if it detected any threats. Remove all items that show up in the scan.