Wood chippings will take nitrogen from the soil, not what you want. Grass clippings are good as long as you don't put more than a couple of inches on at a time so they have time to dry rather than go soggy and slimy. If you can cultivate between the rows a bit, with a hoe, a spade or a rotavator if you have one and there's room, that will loosen up some more soil for earthing up. I have found it's best to leave a good distance between rows to give enough soil for earthing up, but if space is a problem then that can't be done.
I have noticed that some varieties grow closer to the surface than others so for future years it might be worth going for those which grow deep so are less likely to turn green.