During a clap (or clapping) push up you perform the regular push up movement with such force your body goes airborne. Once in the air clap your hands (which is really just for show) before hitting the ground, absorbing the impact forces and exploding upwards again.
I used to think this was a great exercise for building strength and power in the upper body.
Not any more.
In my revised opinion any potential strength and power benefits are comfortably outweighed by the heightened risk of injury this exercise presents.
Wear And Tear
During explosive exercises like the clap push up your joints and connective tissues are exposed to much larger fluctuations in force as you take off and land. Add lots of repetitions and you can easily end up with sore wrists, elbows and shoulders which can take a long time to heal.
Fatigue sets in quickly during clap push ups resulting in an equally quick degradation of exercise technique, particularly when landing. This adds to the wear and tear above and hastens the journey to injury and setback.
For the average trainee who just wants to stay strong and healthy overloading regular push ups with weighted vests and/or resistance bands will be a much better (and healthier) addition to their bodyweight workout routine.
Being strong enough to perform a clap push up is cool... just keep it as a party trick.