We've moved three times in the past year and a bit, and our poor washing machine ended up with a lot of scratches on its paint:
This is a simple problem to correct with an acrylic lacquer spray paint. We've used Dulux Spraypak Appliance White - an acrylic lacquer especially designed for touching up scratches on fridges, washing machines, dishwashers, laundry tubs, or any other white appliance.
Its easy to use. First, clean the area to remove all dust and dirt. Then spray onto the scratches in the same method that you'd use any spray paint. If there are any areas that you don't want spray painted, simply use marking tape to protect these areas, or for larger areas I just use newspaper taped onto the area to protect it from paint.
Be very careful to only use a light spray or you'll end up with a gluggy mess. Its easier to do it lightly and apply more later, than apply too much and have to clean off gluggy paint. Once it was thoroughly dry, I sprayed a tiny bit more onto some of the really bad scratches that hadn't quite been covered. And here is the finished result: