"Mám nějaký problém s autem."

Translation: I have some problem with my car.

[Praha2017]

How should I know that it was my car? It really could have been anyone's car here.
"I have some problem with the car" should be accepted in my opinion.

[VladaFu]

Several people report that but I can't find any example of such use anywhere.

[BoneheadBass]

The translation above is fine, but "...a problem with the car" would be very commonly used, at least in the US East Coast/Mid-Atlantic region. In most cases, the meaning would be understood to be "with my car," though it could easily be used in the sense of, say, "I have a problem with the car that I rented for this trip, and I'm not happy!"

If there's nothing in the Czech sentence that would make "I have some problem with the car" completely wrong, I will add it.

[VladaFu]

I will add it then. I was just surprised I can't find any "problem with the car" (in parenthese) with Google, but that can be a Google's problem.

[wjhira]

The use of "some" is plural, so it needs to say "problems". Either it is a problem singular with the car or some problems with the car.

[BoneheadBass]

The Czech sentence is clearly singular. And, at least in the US, a construction like "I have some problem with my car" is an accepted use. It may help to think of it as "I have some (kind of) problem with my car." And if it makes you feel any better, "I have A problem with my car" is also accepted.