No, not easily. To drive across the border you need "Dual Plates", basically a China license plate bolted below the Hong Kong license plate. To get a China plate requires jumping through a lot of hoops, including proof that you own a business in China, have a certain amount of revenue, etc. etc. Dual plates are not uncommon, but the average motorist will not be able to get access to dual plates.
Occasionally, car clubs will organize group drives into China with special license arrangements. I've never done them though - drivers in China are absolutely nuts. Every single time I've been in China (no exaggeration) I've seen at least one, sometimes two, major accidents. A car I was in nearly rear-ended a BMW 3 series at 120 kph on the highway because the driver decided to come to a dead stop to chat on her phone, not on the shoulder, but in the passing lane. Nutty stuff like this happens all the time. Even worse are the carjackings that happen in Shenzhen (the border town in China adjacent to HK).