If it hurts to put weight into your hands and wrists (as in push-ups or planks), there are some simple adjustments you can make.
Do the exercise on elbows. To maintain body alignment, put blocks or pads under your elbows so that your torso maintains the same distance it would if your hands and forearms were under you.
Put your hands on dumbbells. This keeps your wrists straight. Your forearms and wrists must be strong enough to handle this.
Use your fists on top of padding. Your fingers and wrists must be strong enough to do this safely and comfortably.
Regardless of which modification you use, make sure you train the flexibility and strength of your wrists with other exercises. Doing wrist exercises during your warm-ups, cool-downs, and/or throughout the day will make a big difference to the resilience of your wrists.
Wrist Exercise Videos