  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
$\phi(H)>6$.
