is there a way to get apple care with a phone that is not working properly?

No. You can't. Purchasing AppleCare+ will require a physical inspection of your phone. Since it is ALREADY BROKEN, you will NOT be able to purchase coverage.

Like I said, that's like trying to take out a life insurance policy on someone who is already dead... It's called fraud.

You will have to pay for the repair. IF the phone is less than 60 days old, you may be able to purchase coverage after you pay to have your phone repaired or replaced.

What do you mean "get AppleCare"?

If you're asking how to purchase AppleCare+, you can't. You can't buy coverage for a device that's already broken.

That would be like trying to take out a life insurance policy on someone who is already dead.

Sorry. Cracked = broken.

You can not purchase AppleCare+ coverage for that phone at all.