Let $C_{l}$ be a cycle with length $l.$ The $r$-uniform hypercycle with length $l$ is obtained by adding $r-2$ new vertices in every edge of $C_{l},$ denoted by $C_l^{(r)}$. In this paper, we deduce some higher-order traces for the adjacent tensor of $C_l^{(r)}$ by BEST Theorem. Then we obtain higher-order spectral moments according to the relationship between eigenvalues of power hypergraphs and eigenvalues of signed graphs. Finally, the general expression of the characteristic polynomials of $C_l^{(r)}$ is given. Furthermore, by using this general expression, we present the characteristic polynomials of $C_5^{(r)}$ and $C_6^{(r)}$ as examples.