Learn Finnish: Hän ajaa kilpa-autoa radalla, mutta äiti on huolissaan. - He drives a race car on the track, but mother is worried.

This is a perfect example of a subject-bound state adverbial! It comes from the noun huoli (worry). It is put into the plural inessive case to make huolissa- (in the worries), and then the third-person possessive suffix -an is added. Literally, it means she is "in her worries".

Does the possessive suffix in huolissaan refer to the driver (hän) or the mother (äiti)?

It refers to the mother. With state adverbials, the possessive suffix must always match the subject of the clause it appears in. Since äiti is the subject of the second half of the sentence, the suffix refers to her own state of being worried.

Why is the race car (kilpa-autoa) in the partitive case?

The verb ajaa (to drive) typically takes a partitive object when it describes the ongoing activity of driving a vehicle. Because he is just driving around on the track rather than completing a trip to a specific destination, the object remains in the partitive.

Why do we use the adessive case (radalla) instead of the inessive (radassa)?

The word rata (track or path) refers to an open surface. In Finnish, you generally use the adessive case (-lla/-llä, "on") for open outdoor areas like tracks, streets, and yards, whereas the inessive (-ssa/-ssä, "in") is reserved for enclosed spaces.