A two-layer quasi-geostrophic model for geophysical flows is studied, with the upper layer being perturbed by additive noise. This model is popular in the geosciences, for instance to study the effects of a stochastic wind forcing on the ocean. A rigorous mathematical analysis however meets with the challenge that the noise configuration is spatially degenerate as the stochastic forcing acts only on the top layer. Exponential convergence of solutions laws is established, implying a spectral gap of the associated Markov semigroup on a space of Hölder continuous functions. Moreover, response theory with respect to changes in the average wind forcing is established. Specifically, it is shown that the averages of a class of observables against the invariant measure are differentiable (linear response) and locally Hölder continuous (fractional response) as functions of a deterministic additive forcing. In doing so, a framework suitable to establish (linear and fractional) response for a class of nonlinear stochastic partial differential equations is provided.