System problems can lead to a failure to register or install CCC's helper tool

CCC 7.1 (and later)

When the CCC application is moved, macOS's helper tool registration system will place the helper in a "penalty box". Typically CCC can address the situation by simply unregistering and re-registering the helper tool (without intervention). We have seen a handful of cases, however, where all of CCC's attempts to free its helper tool from the "penalty box" fail, and the user sees a cycle of requests to re-register the helper tool that have no apparent effect. The underlying issue appears to be an invalid code-signing cache entry stored by macOS. If you're finding that behavior, you can try the following steps to correct the problem.

Move the CCC application file to the Trash, then empty the Trash (this will not affect your backup tasks; search for "Carbon Copy Cloner" via Spotlight to verify that no other copies are lingering)
Open System Settings > General > Login Items & Extensions – if all copies of CCC are gone, then the "Carbon Copy Cloner" item should be removed from the "Allow in the background" section
Shut down the computer
Press and hold the Power button until "Loading startup options" appears
Click on "Macintosh HD" to select that volume as the startup disk
Hold down the Shift key and click the "Continue in Safe Mode" button
Log in, then restart the computer again (nothing special, ordinary reboot)
Re-download CCC from here

The brief boot into Safe Boot mode clears out some caches, including the macOS code signature caches. When you open CCC again, you should be prompted to register CCC's helper tool again, and this time it should work normally.

CCC 7.0 through CCC 7.0.4

If you are encountering problems with installing CCC's helper tool and you are using a version of CCC that is older than 7.1, we recommend that you apply the 7.1 update prior to attempting any of the troubleshooting suggestions below.

Configuration files for privileged helper tools are placed in the /Library/LaunchDaemons folder on your startup disk. CCC never touches this folder directly, rather it uses the macOS "Service Management" service to install and load its helper tool configuration. If the permissions or ownership of this folder are incorrect, however, the Service Management daemon (smd) will fail to install the helper tool configuration, and this service offers no recourse. Often the helper tool installation will fail with a nondescript error, e.g. "CFErrorDomainLaunchd error 2". In most cases, reinstalling macOS does not repair the affected system folders. We have reported this system problem to Apple (FB11188842) and we are currently waiting for a response, but there are a handful of options that you can leverage to resolve this permissions problem.

Preliminary troubleshooting of helper tool installation failures

Reboot

The first troubleshooting step is always "Reboot your Mac". After rebooting, open CCC to see if you are still prompted to load CCC's helper tool.

Toggle the CCC background item

If the problem persists after rebooting:

Quit CCC
Open the System Settings applications
Navigate to General > Login Items
Toggle the switch next to "Carbon Copy Cloner" (On, or if it's already on, toggle it off, then back on).
Open CCC to see if you are still prompted to load CCC's helper tool

Force-load the helper tool via the legacy system interface

If the problem still persists, then you can try forcefully loading CCC's helper tool in case some application used an older macOS interface to disable it. Paste the following into the Terminal application:

sudo launchctl bootstrap system /Library/LaunchDaemons/com.bombich.ccchelper.plist

sudo launchctl enable system/com.bombich.ccchelper

Press the Return key after pasting in each line, then authenticate when prompted. Afterwards, try again to open CCC and save/run a backup task.

Advanced troubleshooting when more serious system problems are present

Remove the contents of the affected folders, then correct their ownership and permissions

If the problem persists after trying the steps above, then the next troubleshooting step is to remove the affected system folders and recreate them with the correct ownership and permissions. To avoid exposing yourself to potential security vulnerabilities, it is imperative that you remove the content of these folders rather than simply correcting the ownership and permissions. Make a note of any applications listed in these folders – those applications should later be re-opened so they have an opportunity to reinstall their helper tools.

Paste the following into the Terminal one line at a time. Press the Return key at the end of each line, authenticate when prompted:

cd /Library
sudo rm LaunchDaemons/*
sudo rm PrivilegedHelperTools/*
sudo chown root:wheel LaunchDaemons
sudo chmod 755 LaunchDaemons
sudo chown root:wheel PrivilegedHelperTools
sudo chmod 1755 PrivilegedHelperTools

If any of these commands produces an "Operation not permitted" error, or if you are still unable to save a task in CCC, then proceed to the next section.

Replace the folders via Recovery Mode

If macOS security and privacy restrictions prevent you from correcting the issue while booted from your Mac's production startup disk, you can perform the tasks in the Terminal application while your Mac is booted in Recovery mode.

Intel Macs: Hold down Command+R while rebooting. Apple Silicon Macs: Shut down, hold down the Power button until the startup options appear, then select Options.
Choose Terminal from the Utilities menu in the menubar.
Type the following into the Terminal one line at a time, pressing the Return key at the end of each line:

cd "/Volumes/Macintosh HD/Library"
rm -rf LaunchDaemons
rm -rf PrivilegedHelperTools
mkdir LaunchDaemons
chown root:wheel LaunchDaemons
chmod 755 LaunchDaemons
mkdir PrivilegedHelperTools
chown root:wheel PrivilegedHelperTools
chmod 1755 PrivilegedHelperTools

Note: If your production startup disk's name is not "Macintosh HD", substitute the correct name in the first line above.

After you have completed those steps, reboot your Mac, open CCC, and try again to save or run a backup task.