If you're running it of a mini itx computer that is built into your vehicle then you may still have power issues. You may try getting a cigarette lighter to usb adapter and connecting the thinner cable on your Y splitter to that. Just off the top of my head though, I don't know a great deal about car computers.
You could test that theory without spending any money would be to connect the same USB connector into a laptop and trying it with the car running. A laptop will easily power your drive buy itself so it will let you know if your issue is power or not. It seems odd if it is, but as i say i don't know much about car computers..
The thinner cable on the Y is only a power supply, so you shouldn't have any trouble splitting the source.. If it is a permenant solution though, you could hardwire a usb connector into your vehicle. That would give you back the lighter (if you use it) and allow you to install a much more appropriate fuse to protect your HDD.