In this work, we derive a new water-wave model by using an ansatz for the velocity potential inspired by shallow-water theory. The velocity potential is represented by a series with unknown functional coefficients that dependent on the horizontal variable and time and vertical functions (polynomials) that match the ones appearing in an asymptotic expansion. We show that the derived equations have a canonical non-local Hamiltonian structure in accordance with the Hamiltonian formulation of the full wate-wave problem. We discuss the relation with existing models and provide some numerical results.