Let $A$ be an abelian variety and $X$ a cone over $A$. Then $X$ is log canonical, but as soon as $\dim A\geq 2$, then $X$ is not Cohen-Macaulay. (This you can see by computing the local cohomology at the vertex).
For the $c=1$ case: those are obviously CM. Otherwise one can study the codimensions of various embeddings of abelian varieties.