I would not use a small piece of sapphire to embed diamond in a Batt lap. It may score the lap. Apply the Diastik lightly or use a clean finger to spread the diamond thinly, and grind or polish away. When black swarf builds up on the lap or the stone, wipe the lap with a piece of paper towel dampened with WD-40, and reapply some diamond – sparingly. Clean the stone and dop with alcohol on paper towel before polishing. I give Batt laps a cleaning wipe after use. Sometimes I clean the splash pan before polishing, but not always, and I avoid touching it. After cutting a few stones I clean the machine as best I can, but without being obsessive. The Matrix laps are also given a wipe with dampened paper towel to clean them. If you are going from an oil-based diamond to a water-based oxide, or vice versa, on the same lap wash it with water and some household detergent. (I use separate laps for oxides and diamond polish.) Most of my Gearloose laps live in the original cardboard covers they came in, just with a clean piece of paper towel on the working surface to protect it.
The main thing is to use minimal diamond or oxide when polishing. If you get scratches it usually means the lap needs a wipe to remove excess polishing medium. If scratching persists change polishing direction through 90 degrees. Check your prepolish. Has it removed all previous grooves and pits? Clean the polishing lap thoroughly. If it is a Matrix+oxide lap scrub it with water and detergent. If it is a diamond lap wash it, dry it, and apply a synthetic rubber eraser to the lap running in reverse. Try different lap and polish combinations. If all that doesn’t work try a different stone. Resurfacing a lap is the very last resort.