In my understanding, both 片方の手 and 片手 mean "one hand". What is the difference and how to use them correctly?
Here are the first 10 hits for 片方の手 in BCCWJ Corpus:
As you can see, and as goldhick "answered" as a comment, 片方の手 tends to be used when it is contrasted with the other hand. (ie, "One hand ～, the other hand ～")
片手 is simpler; "one hand", "one-handed", etc. When you say 片手で運転する, what the other hand is doing is not important.
Another example: 岩を片手で持ち上げた simply means someone lifted a rock with one hand. 岩を片方の手で持ち上げた is almost the same, but it may sound like he was doing something else with his remaining hand.