It's called Autotesting and, providing you have a competition licence and are a member of a recognised motor club, you can enter in your standard road-going car. It's a bit tough on the gearbox and clutch though.
Many years ago when I was actively involved in motorsports I competed in an MG Midget. It was also hard on half-shafts and I frequently had to replace them. A demon tweek was to replace the rear wheels with 12" wheels off a Hillman Imp just to encourage the back end to lose traction.
Later on I was a Clerk of the Course on a few Autotests that my club promoted. That was when all my "spare" money had to go into the house and I couldn't afford to compete.
Happy days