HELP!!!! Diagnosis this problem.
 

So I have an issue that I need help with.
My 2005.5 2.0 turbo is having an issue with dying out. The car will run with what seems to be a little misfire...At least if I push it hard. Anyways, what will happen is it runs and idles fine and then just dies out. After it dies the car becomes to start. In fact it won't start at all... Meaning it will turn but not fire. But if it sits for a while... Like over night.... It will start up… let it sit for a few minutes and it will start to turn and fire but not catch. The fuel pump in the rear sounds to be buzzing once the key is turned. It does stop after maybe 30 seconds.
But like I said, once it does turn over , fire, catch, and run... It does so fine... Just with a little misfire.

The codes I was able to get off my ELM327 and a my Torque and Carista app on my phone are:
P0089
P0304

If any one could please help me diagnosis this problem and tell me where the issue is and maybe how to fix it.
Thank you so much

It's probably your low pressure fuel pump (the one in the tank), but you also need to pull the HPFP and check the cam follower.

If you have access to a VAGCOM, you can watch the duty cycle of the LPFP. Anything over 50% at idle and 80% at load is too high and a sign it's failing. That's where I'd start.