Catalina endless installation

You will need a copy of the Sierra installer. If you don't have a copy, then you can try downloading one here: How to upgrade to macOS Sierra. Alternatively, you can try one of these:

MacOS Sierra - Pick one

https://itunes.apple.com/app/macos-sierra/id1127487414?mt=12

https://apps.apple.com/app/id1127487414?mt=12

With a copy of the Sierra installer you should make a bootable USB flash drive installer.

Bootable USB Installers for OS X Mavericks, Yosemite, El Capitan, and Sierra

First, review this introductory article: Create a bootable installer for macOS. Second, read this How To outline for creating a bootable macOS installer. Simply use the Terminal command by copying and pasting the command line for the version of macOS from the list below. You will need an 8GB or larger USB flash drive that has been partitioned and formatted for use with OS X as well as an installer for the desired macOS version that you have previously downloaded from the App Store.

Drive Partition and Format

1. Open Disk Utility in the Utilities' folder.
2. After Disk Utility loads select the drive (out-dented entry with the w/type and size info) from the side list.
3. Click on the Erase button in the Disk Utility toolbar. Name the drive, "MyVolume". <---- IMPORTANT!
4. In the drop down panel set the partition scheme to GUID. Set the Format type to Mac OS Extended (Journaled.)
5. Click on the Apply button and wait for the Done button to activate. When it does click on it.
6. Quit Disk Utility.

Create Installer

Open the Terminal in the Utilities' folder. Choose the appropriate command line (begins with "sudo") depending upon what OS X installer you want. Paste that entire command line from below at the Terminal's prompt:

Command for macOS Catalina:

sudo /Applications/Install\ macOS\ Catalina.app/Contents/Resources/createinstallmedia --volume /Volumes/MyVolume

Command for macOS Mojave:

sudo /Applications/Install\ macOS\ Mojave.app/Contents/Resources/createinstallmedia --volume /Volumes/MyVolume

Command for macOS High Sierra:

sudo /Applications/Install\ macOS\ High\ Sierra.app/Contents/Resources/createinstallmedia --volume /Volumes/MyVolume

Command for macOS Sierra:

sudo /Applications/Install\ macOS\ Sierra.app/Contents/Resources/createinstallmedia --volume /Volumes/MyVolume --applicationpath /Applications/Install\ macOS\ Sierra.app

Command for El Capitan:

sudo /Applications/Install\ OS\ X\ El\ Capitan.app/Contents/Resources/createinstallmedia --volume /Volumes/MyVolume --applicationpath /Applications/Install\ OS\ X\ El\ Capitan.app

Command for Yosemite:

sudo /Applications/Install\ OS\ X\ Yosemite.app/Contents/Resources/createinstallmedia --volume /Volumes/MyVolume --applicationpath /Applications/Install\ OS\ X\ Yosemite.app

Command for Mavericks:

sudo /Applications/Install\ OS\ X\ Mavericks.app/Contents/Resources/createinstallmedia --volume /Volumes/MyVolume --applicationpath /Applications/Install\ OS\ X\ Mavericks.app

Press RETURN. You will be asked for your admin password. It will not echo to the Terminal window. Then press RETURN again. Wait for the return of the Terminal prompt signifying the process has completed. It takes quite some time to finish. Be patient.

The bootable installer can be used to install the selected version of macOS. Shut the computer down. Insert your flash drive into a USB port. Start the computer like so:

Boot Using OPTION key

1. Restart the computer.
2. Immediately after the chime press and hold down the  "OPTION" key.
3. Release the key when the Boot Manager screen appears.
4. Select the disk icon for the USB flash drive.
5. Click on the arrow button under the disk icon.

The installation has failed due to a problem either with the drive or your old system. I assume you are using a different computer. Is that correct or have you been able to regain control over the Mini? Please describe how you were upgrading to Catalina. Have you tried booting into safe mode?

Boot Into Safe Mode

1. If your Mac isn’t already shut down, then shut it down and wait at least 30 seconds before rebooting.
2. Immediately, at or before the chime, press and hold down the SHIFT key. 
3. Release the key after the Apple logo and progress bar appear.
4. Safe mode startup is much slower than normal startup, so be patient.
5. When the Login Screen appears enter your admin password. If you use automatic login, then this means you are in safe mode.

Before You Upgrade macOS

1. Fix any problems before upgrading.
2. Run First Aid in Disk Utility while booted from your Recovery HD.
3. Backup. Make a bootable clone using CarbonCopy Cloner or use TimeMachine.
4. Be sure your startup disk has 18-20GB of free space during the installation.
5. Upgrade third-party software for compatibility. See App Compatibility Table - RoaringApps. If you have mission critical software, then DO NOT upgrade until you know that software is compatible with the version of macOS you are about to install.
6. Disconnect all third-party peripherals.
7. Download the installer which will be in the Applications folder. Quit the installer after it opens, then make a copy of the installer and save it in the Downloads folder. The one in the Applications folder will be deleted after a successful installation.
8. The download is large, anywhere between 5 and 9GBs. It takes time to complete. Be patient.
9. Double-click on the installer in the Applications folder. Click on the start button to begin the installation.

I think so. There is an incomplete installation of Catalina that it is trying to complete. If it cannot complete the installation, then you will have to erase the drive and restore from a backup or try using Internet Recovery:

Internet/Network Recovery of El Capitan or Later on a Clean Disk

     If possible, back up your files before proceeding.

You will need a fast, working Internet connection, preferably, Ethernet. If you must use Wi-Fi, you will be asked to select your network and enter the network password. This procedure will install the current version of macOS that is compatible with your model.

1. Restart the computer. Immediately, at or before the chime, hold down the (Command-Option-R) keys until a globe appears.
2. The Utility Menu will appear in from 5-20 minutes. Be patient.
3. Select Disk Utility and click on the Continue button.
4. When Disk Utility loads select the target drive (out-dented entry w/type and size) from the side list.
5. Click on the Erase button in Disk Utility's toolbar. A panel will drop down.
6. Set the partition scheme to GUID.
7. Set the Format type to APFS (SSDs, only if installing Mojave or Catalina ) or Mac OS Extended, (Journaled.)
8. Provide a volume name, usually "Macintosh HD."
9. Click on the Apply button, then click on the Done button when it activates.
10. Quit Disk Utility and return to the Utility Menu.
11. Select Install OS X and click on the Continue button.