A 宜しい (good, okay, all right) request is when you send your best regards to someone. This is a set phrase you say when someone is going to help you out and you're all like "thank you for your help" or "I'm counting on you" or some combination of the two. It's a word you'll see a lot to varying degrees of kanji and length.
The readings come from 宜しい and お願いします.
Why do Japanese people say "please treat me well" all the time?