How can I restore a PS4 following a hard shutdown (unplug of power)?

It sounds like something has become corrupted - your best bet is to try the various 'factory reset' options offered by the PS4's Safe Mode.

To boot to Safe mode, hold down the power button on the console until you hear two beeps. It should take about 8 seconds.

Once you get to Safe Mode, you will get a menu of options. The ones we're interested in are below, you should try them in the order they are presented:

4. Restore Default Settings - this will reset everything on the PS4 to its original software settings, but keep your data (profiles, save files etc). Try this first.
5. Rebuild Database - while this is more aimed at improving boot times, a little-known fact is that it will remove games and applications from the database that it thinks are 'corrupted'. Running this can take a few hours though.
6. Initialise PS4 - this will wipe your hard drive (removing downloaded games, applications, save files etc) and restore you to factory settings. The only thing it won't do, is touch the PS4's operating system (system software). You have a choice of doing a 'Quick' or a 'Full' initialisation. Try the Quick one first, then if that doesn't work, move to the Full re-initialisation.
7. Initialise PS4 (Reinstall system software) - This is a full wipe of everything, but requires you to download the PS4 system software onto an external device.
   • Get the 'PS4 reinstallation file' here: https://www.playstation.com/en-au/support/hardware/ps4/system-software/. Note you need the reinstallation file, not the update file
   • Get a USB thumbdrive that uses the FAT32 drive format, or format one yourself.
   • Create a folder named PS4 on the USB drive
   • Inside that folder, create another folder named UPDATE.
   • Copy the installation file and save it in the UPDATE folder. Save the file as PS4UPDATE.PUP.
   • Plug the USB drive into the PS4
   • Start the PS4 console in Safe Mode
   • Select Safe Mode option 7: Initialise PS4 (Reinstall System Software).
   • Select Update from USB Storage Device -> OK.

If none of the above options work, then your PS4 is well and truly broken, and you should take it to a computer repair shop to see if they can help. I doubt pulling the plug would be covered by warranty.

Sources

• Personal Experience
• PS4 Reinstallation Instructions - Playstation Support
• 'How to factory reset a PS4' - Digital Trends