An analogue of the Mukai map m_g: {mathcal {P}}_g rightarrow {mathcal {M}}_g is studied for the moduli {mathcal {R}}_{g, ell } of genus g curves C with a level ell structure. Let {mathcal {P}}^{perp }_{g, ell } be the moduli space of 4-tuples (S, {mathcal {L}}, {mathcal {E}}, C) so that (S, {mathcal {L}}) is a polarized K3 surface of genus g, {mathcal {E}} is orthogonal to {mathcal {L}} in {{,mathrm{Pic},}}S and defines a standard degree ell K3 cyclic cover of S, C in vert {mathcal {L}} vert . We say that (S, {mathcal {L}}, {mathcal {E}}) is a level ell K3 surface. These exist for ell le 8 and their families are known. We define a level ell Mukai map r_{g, ell }: {mathcal {P}}^{perp }_{g, ell } rightarrow {mathcal {R}}_{g, ell }, induced by the assignment of (S, {mathcal {L}}, {mathcal {E}}, C) to (C, {mathcal {E}} otimes {mathcal {O}}_C). We investigate a curious possible analogy between m_g and r_{g, ell }, that is, the failure of the maximal rank of r_{g, ell } for g = g_{ell } pm 1, where g_{ell } is the value of g such that dim {mathcal {P}}^{perp }_{g, ell } = dim {mathcal {R}}_{g,ell }. This is proven here for ell = 3. As a related open problem we discuss Fano threefolds whose hyperplane sections are level ell K3 surfaces and their classification.