When I think about this question, I found the cut locus of torus is the combine of some curves of generator. As picture below, the cut locus of $p$ is like the black circles. Also the black circles are the generator of fundamental group of torus. In fact, I am sure it is right for all orientable surfaces if assume the generator of trival group is a point. Because the all orientable surfaces is n-torus or $S^2$. The cut locus of them is the generator of fundamental group.
But for the high dimension or non-orientable, I don't knwo whether it is right, so , anybody know some theorem about this ? Thanks.
What we know is that for all $x\in M$, the cut locus $c(x)$ is a closed measure zero set in $M$, and $M\setminus \{x\}$ deformation retracts onto $c(x)$. This introduce a topological constraint on the cut locus. For example, the inclusion
$$i:c(x) \to M$$
induces isomrphism $i_* : \pi_k (c(x)) \to \pi_k (M\setminus \{x\})$. That's why you see the two generators of $\pi_1 (\text{torus})$ in the cut locus in your example.