Gauss-Prym maps on Enriques surfaces

We prove that the $k$-th Gaussian map $\gamma^k_{H}$ is surjective on a polarized unnodal Enriques surface $(S, H)$ with $\phi(H)>2k+4$. In particular, as a consequence, when $\phi(H)>4(k+2)$, we obtain the surjectivity of the $k$-th Gauss-Prym map $\gamma^k_{\omega_C\otimes\alpha}$ on smooth hyperplane sections $C\in \vert H\vert.$ In case $k=1$ it is sufficient to ask $\phi(H)>6$.