"Asteroid Reprocessing" is the recipe which allows you to put in a chunk of a given type (say, metallic) and receive as an output:
40% of the time, another metallic
20% of the time, a carbonic
20% of the time, an oxide
and 20% of the time, nothing
When combined with quality modules, you basically get to roll a quality chance and still get an asteroid back 80% of the time. Many people (including the devs) are critical of this behavior, as they believe it trivializes quality.
Asteroid processing (notice, processing and not REprocessing) refers to the recipes which turn an asteroid into ingredients. The basic ones have a 20% chance to return the asteroid you put into it. This recipe is affected by productivity, which means, at Asteroid productivity 30 (which is an infinite research you can do), you attain 300% additional productivity, so 4x the total product, which means your return rate for that specific asteroid type again becomes 80%. This means you can still use asteroid cycling for quality, it just scales based on your asteroid productivity research, and requires a bit of additional logistics to deal with the actual ingredients produced by the recipe.
Hope that helps, let me know if you have any other questions.