Do I have a case if a plate shattered, sending little pieces of the plate into our food and drinks and cut the top layer of skin off my finger?

My boyfriend and I went to a restaurant to eat dinner. We got our drinks and some of our food. While waiting, a waitress dropped a plate and cut the skin off my finger. It was bleeding pretty badly for a while. The manager got me a bandage and some cream. The combed our meal, but he did not fill out an accident report. As soon as we got home, we called and asked him why he did not file one out, he said that he didn't think it was that serious and that we got our meals for free. We asked him to fill one out, and he said their insurance company will be contacting me in a couple days. I didn’t need medical attrition, however, this is going to scar my finger, and there is a pretty good chunk taken out. I am not sure what to do, or even what to say to the insurance company.