Your arms and hands are two of five contact points between you and the bike. On bumpy roads, it's critical that the mass of your body is supported independently from the bike. Try to maintain a relaxed posture with slightly bent elbows. When the road gets bumpy, take weight off your saddle by transferring a small amount of body weight to your feet and hands. This might not feel natural at first, but with time, your elbow and shoulder joints will thank you with fewer stress-related injuries. You'll also be able to absorb those sudden unexpected bumps more easily.
Don't make the mistake of staying in the saddle when you encounter a dip in the road. This is a common mistake inexperienced riders make while racing criteriums. The corners on criteriums are also the spot where engineers design dips in the road for water runoff, which makes it easy to be bucked off the saddle when the rear wheel drops in and out of the dip. If you're lucky, you can maintain control and stay upright. Sometimes the rear wheel pops up and swings to the outside due to the force of the turn. When your tire impacts the road again, you can lose traction and crash.
The next time you're at a race, watch how experienced riders enter corners and see if their bikes dip. If the dip is significant, notice how racers will raise-up slightly off their saddle with bent elbows and allow the bike to drop down into the dip while keeping their upper body at a constant level. Their arms and legs serve as shock absorbers to allow the bicycle to move up and down following the road.
Your body is the largest mass in the whole human-machine equation. If it remains stable through the turn, you can always control the bike.