Prius making terrible noise

Are there warning lights on the dash? If so they have stored error codes that will let you know what the problem is.

 

Without codes I would suspect a head gasket leaking coolant into a cylinder. Usually they will run ok once it is warms up unless its really bad. If its losing coolant or pumping out white smoke, a head gasket is very likely. Other misfire problems can cause the same issue and can be plugs, coils or injectors. However these problems generally will not go away as fast and will definitely set codes.

The reason it sounds bad is the engine is not achieving a smooth idle rpm fast enough to sync with the transaxle motors. This causes a clutch like damper to slip making the bad racket. So it could be a startup misfire caused by plugs, coils or injectors. A headgasket leak can foul a plug which may clear fast when running.

 

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How many miles are on the vehicle?

Have you already tried using the oil dip stick to check the engine oil level lately?