There were reports of this happening when the game first released. It was just a case of waiting for the game to fully download all updates, restart PS4, plug in the wheel, then start the game.
I have not heard of a disc being faulty? Let us know so we can give advice in the future.
Edit: Here is the answer from WMD.
http://forum.projectcarsgame.com/showthread.php?22489-CAREER-LOCKED-Only-can-play-weekend-race-PS4
You FIRST need to create a DRIVER, then that will unlock the CAREER. Don't do ONLINE or QUICK RACE first. CREATE DRIVER first.