The phrase "clean knife" means a knife that doesn't have any dirt, food, or other substances on it. It's a way to describe a knife that is free from any mess or residue. Full definition
Cut pie slices, cleaning the knife in between each cut, to get a clean nice sharp slice, and serve with a generous spoonful of the cinnamon whipped cream. (yes-moreplease.com)
With a warm clean knife, slice square, rectangular or diamond shapes. (angelaskitchen.com)
As you can see I didn't do such a great job on cleaning the knife after each slice. (thecandidappetite.com)