This is what the Apple Support article advises for your errors:

iOS: Restore errors 4005, 4013, and 4014

When you try to update or restore your iPhone, iPad, or iPod touch, you may see one of these messages in iTunes.

You may see one of these messages if the device unexpectedly disconnects during the update or restore or iTunes couldn't tell the device to go to the restore state.

• The iPhone [device name] could not be restored. An unknown error occurred (4005).
• The iPhone [device name] could not be restored. An unknown error occurred (4013).
• The iPhone [device name] could not be restored. An unknown error occurred (4014).

Try these steps to resolve the issue:

1. Install the latest version of iTunes.
2. Restart your computer.
3. Make sure your computer is up to date. If an update requires a restart, check for updates again after you restart.
   
   • Learn more about updating OS X.
   • Learn more about updating Windows.
4. Restore using another USB cable.
5. Restore your device on another computer.

If you continue to see error 4005, 4013, or 4014 when you restore your device, contact Apple for support.

Learn more about other update and restore errors.

Here is a link to the Error Messages List:

Get help with iOS update and restore errors - Apple Support

Best of luck,

GB