We have a few things that are go-to for stains.
Obviously our spot gun gets the most use.
Krud Kutter is great for non-ink stains. We use it a lot on stuff like this because it doesn't damage fabrics. It's especially good on light grease and oil stains.
Tide pens are fantastic for things that krud kutter won't get. I don't mix the two. I don't know what would happen, but with small dots on embroidery from the machine, tide pens work wonders. The stains usually disappear without any extra cleaning.
I hope this helps. I with those small dots, try a tide pen.