It will depend on the terms of your fully comp (comprehensive) auto insurance policy in the UK if your vehicle would be covered for damages sustained in a single car accident, such as you falling asleep at the wheel and crashing your vehicle, however in general this damage should be covered.
In the UK, fully comp should cover for accidental damages your car sustains if you are at fault in an accident. If you fell asleep at the wheel and damaged your vehicle then this should be accidental damages since obviously you did not intentionally fall asleep and wreck your vehicle.
With Fully Comprehensive insurance your car is covered in case of accidental damage on your part. Accidental damage is that done to the insured's vehicle including malicious damage, and windscreen damage. Some insurance providers' comprehensive policies also include extra coverage things such as personal accident cover, European driving cover and courtesy cars.
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[Let CarInsurance.com help you find affordable auto insurance now.] Many comprehensive car insurance policies also come with added extras such as windscreen cover and the ability to drive other cars, although you should never assume that you have these additions, always check with your auto insurance broker / agent.
The exact coverage a Fully Comprehensive (also known as fully comp or comp) policy has attached to it can differ depending upon what specific coverages an insurance company will offer so if you have damages to your vehicle for which you would like to make a claim for, contact your insurance agent to find out for certain if your fully comp coverage covers your accident and if so how to make your first party claim. | 
