When dropping out of the air and going full throttle (after a power loop or something before hitting the ground), what causes the wobble on the quad? What can be done to prevent this?

Thanks

I think what you describe is "propwash", meaning the quad falls into its own prop turbulences. There's multiple things that have influence on how noticeable this "wobble" is:

• technical aspect: the stronger the motors & escs & batteries / the lighter the props / the "stiffer" the PIDs => the faster the quad can react to the quickly changing conditions within the disturbed air. Also, too much filtering (at too low frequencies) will lead to a delay between the real gyro information and the PID's response, which can have a big influence here.
  
  
• piloting aspect: you can to a certain degree "fly around" propwash - for example, instead of dropping flat and then punching throttle, you can push the nose down and "transition" into forward movement smoothly. Or, when coming out of a powerloop, try not to apply throttle too late, but slowly increase it gradually. And hard 180° turns where the quad comes to a dead stop for a moment imho are just a complete no-go - there's more elegant ways around corners.
  

We can of course go into many details here (don't hesitate to ask), but as a first answer, i hope this helps as a rough overview.

Also referred to as Vortex Ring State.
Here is a video from Quadcopter101 that helped me back when I first started on a Syma X5C