Do faster moving projectiles make larger penetration holes? When it comes to shooting guns, a bullet of a certain size will make a certain size hole in the paper or wood target.  
A $.22$ bullet makes a pea size hole regardless of if its shot out of a pistol and going $300$ m/s  or shot from a rifle and going $800$ m/s.
Does this continue as you increase the speed?  Will a $.22$ bullet going $10,000$ meters a second still make a pea size hole in a paper target?
What if it was going a fraction of $c$? 
Does this change if you exchange the paper target for a metal plate?
 A: That is very interesting question. 
At relatively low speeds (1-2 Mach) and light targets amount of heat generated during impact is negligible, hence bullet just shears piece of target and goes on flying. Heat is dissipated and does not cause additional damage. 
At higher speeds, and metal targets - amount of heat increases dramatically, generated heat is enough to vaporize part of target as well as bullet (even at Mach 2 impacts you can see a flash of light at impact), and shockwave from this vaporized target and bullet rips everything around. 
Another effect I've heard is that during high speed metal impact - electron cloud could be displaced inside the metal and increase the damage, but I cannot find a reference. 
You can see a video of this Mach 7 impact and embrace the amount of heat generated during impact: https://www.youtube.com/watch?v=i737rM6FxqE
So you don't need relativistic speeds to see fun impact effects. 1km/s impact speed already change alot. At 7km/s damage becomes spectacular: https://www.esa.int/spaceinvideos/Videos/2018/02/Hypervelocity_impact_testing . This speed range is studied in the context of shielding of satellites from micrometeorites, which can have up to ~50km/s impact speed if you are lucky.
A: You're asking about terminal ballistics. As a projectile hits a target the initial hole isn't about the speed of the projectile but rather how the projectile is decelerated. So going through a paper target is hardly noticeable. However a mostly lead 0.22 projectile won't have the structural integrity to punch through a "thick" steel plate. Rather the projectile would just fragment. 
Interesting though to leave nice round holes in a paper target does require some careful selection of projectile and target. Target shooters use "wad cutters" which are bullets with a very cylindrical shape. The rather sharp edge helps cut a clean hole. Also good target paper is made using short fibers for the paper. Using long fibers tends to make the target tear rather than have the projectile act as a punch.
Consider shooting a 3/4 inch plank. The greatest difference between different types of 0.22 projectiles and their velocities won't be about the entrance hole, but rather in the exit hole. A hard fast bullet like depleted uranium will make a clean hole in the plank. A slower soft lead bullet would make a small hole in front and a massive exit hole because the soft lead would mushroom inside the plank. 
