Guess there many ways. I not put a lot of effort into teak care.
1. Sand out scratches and splinters with a 400 grit sand paper. Have used finer 600 + grit just to clean it (in a way). If it a bad Gouge or Splinter I may use a 180-220 grit to rough it down to blend/form it with the wood around and then use 400 + to finish.
2. Sweep it off with a Paint Brush or what you want to use. Wash off with clean water to remove sanding dust and let dry.
3. Oil with a teak oil periodically. Depending on my personal situations I not even use oil "sometimes".
Just 1 & 2.