If you have modified your car, in the UK you can purchase performance car insurance, which includes breakdown cover.
For starters, there is a lot of difference between a sports car, which is a high performance vehicle, and a modified one. The sports car could be a big-ticket item like a Merc or a Porsche, while a modified car is one which you have customised to your fancy. The latter is likely to be more expensive to insure or get breakdown cover.
The breakdown cover on your modified vehicle could depend on the insurance policy you have; which could be third party only coverage, third party coverage, fire and theft coverage, or full coverage.
The cost of insurance cover will depend on how old the car is, the type of car and how much it has been modified. Some modifications will weigh in on the policy so before your add the modification please discuss with the insurer.
The cost of insuring performance cars is higher, mainly because statistics show that they are more prone to accidents. Many insurance companies are of the opinion that these modifications will not be good for the car. The cost of the insurance will also depend on the kit and the engine power.
The breakdown cover is an option on the regular insurance cover for your performance vehicle. The level of service you opt for would depend on your personal requirements.