What should I do about a gap between a kitchen backsplash and the cupboards?
The grout lines in the mosaic tile look like 1/8" is that right? 

If that's the case you need to space the tiles up off the counter using 1/8" spacers. They shouldn't be directly on the counter as shown in your picture where you're just holding it. When grouting don't grout that area. Use a silicon caulk that matches your grout instead.

That leaves 1/4"  grout line between the top of the tile and the bottom of the cupboard which isn't ideal but shouldn't really be noticeable at that spot. That covers the 3/8" gap.

Or you can split the difference and do a 3/16" on top and bottom. May even want to try a black caulk for the top so you don't notice it against the black cabinets.