What is a 'safe mode' and how can I use it to potentially fix the reinstallation error?

PS4 Safe Mode allows you to access system functions that bypass the normal operating system. To enter Safe Mode, turn off your PS4 completely, then press and hold the power button until you hear two beeps (the second beep occurs about 7 seconds after the first). From Safe Mode, you can try options like 'Rebuild Database' (which can fix corrupted files) or 'Initialize PS4 (Reinstall System Software)' – be aware that the latter will erase your data.