Tightening abdominal muscles, back muscles and gluteal muscles can better transmit strength and exercise core muscles.
When someone is doing push-ups, their elbows will bounce. On the whole, it's not a big problem, just the sound of muscles and bones at the elbow. You can try to adjust the direction of your elbow.
Attention is to clamp the chest instead of straightening the elbow, which can better train the chest instead of the arm.
In addition to exerting force in the ascending stage, we should also exert force slowly in the descending stage (muscle centrifugal contraction)
I hope my answer can help you.