As you have already backed up your Mac with Time Machine, you can fix the Fusion Drive without losing data, or replace the Fusion Drive and then restore from the Time Machine backup drive.
When Fusion Drive Mac won't boot up, you can try to fix it in these ways:
In case, all the methods above fail to fix the Fusion Drive on your Mac Mini, you may consider replacing the Fusion Drive, SSD is a good choice. You should ensure the new drive's capacity is larger than that of the Time Machine backup.
If you receive the error "This disk does not have enough space to restore your system," the Time Machine drive may report a wrong disk space or the drive is corrupted, you can try to recover files from corrupted Time Machine drive, or run iBoysoft Data Recovery in macOS recovery mode to save files from the Fusion Drive before being overwritten.