We prove that the 𝑘th Gaussian map 𝛾 𝑘 𝐻 is surjective on a polarized unnodal Enriques surface (𝑆, 𝐻) with 𝜑(𝐻) > 2𝑘 + 4. In particular, as a consequence, when 𝜑(𝐻) > 4(𝑘 + 2), we obtain the surjectivity of the 𝑘th Gauss-Prym map 𝛾 𝑘 𝜔 𝐶 ⊗𝛼 , with 𝛼 ∶= 𝜔 𝑆| 𝐶 , on smooth hyperplane sections 𝐶 ∈ |𝐻|. In case 𝑘 = 1, it is sufficient to ask 𝜑(𝐻) > 6.

We prove that the k-th Gaussian map γ k H is surjective on a polarized unnodal Enriques surface (S, H) with ϕ(H) > 2k + 4. In particular, as a consequence, when ϕ(H) > 4(k + 2), we obtain the surjectivity of the k-th Gauss-Prym map γ k ω C ⊗α on smooth hyperplane sections C ∈ |H|. In case k = 1 it is sufficient to ask ϕ(H) > 6.