Hit and Run, often shortened as HnR, is a glitch that allows you to gain horizontal speeds even more powerful than a slingshot.
However, it isn't useful outside of specific categories such as New Game+, since the requirements to perform it are a lot less versatile than slingshot's, and one of these requirements is unlocking Spin Slash, which requires 85 additional skill points.
To perform Hit and Run, you must start a boost right after hitting a spinny boi with the first hit of a spin slash.
It's also possible on Ninja type enemies (Ninja, Shinobi, Kunoichi and Master Ninja), though it's not as useful.
Unlike for slingshots, Sonic's relative position to the enemy, as well as the enemy's rotation, will significantly affect the resulting speed, as well as the timing window, which may range from impossible to 3 frames.
For this reason, each spinny boi requires its own specific setup for the glitch.
The general idea is to double jump, dodge, hold the jump button to initiate a spin slash, then boost away.
When doing Hit and Run, you want to pay attention to the spinny boi's wings, as if you try to go right and hit the left of the wing it will block you (and vice versa), and if you hit the complete center of the spinny boi the glitch won't work.
This video by Programfairy shows how to do Hit and Run, as well as multiple setups on Chaos Island, where it is primarily used:
With some additional setup, it's possible to do Hit and Run on enemies other than the two types shown above.
To do this, you must first throw a single wave of Sonic Boom at the enemy, and then perform Hit and Run at the same time as the Sonic Boom hits.