We consider the question of the density in the space L^p, 1 ≤ p ≤ ∞, on the unit circle, of the subspaces H^p+σ^m_k=1 w_kH^p, where H^p is the standard Hardy space and w₁,...,w_m are given functions in the class L^∞. This question is closely related to problems of uniform and L^p-approximations of functions by polyanalytic polynomials on the boundaries of simple connected domains in ℂ. The obtained results are formulated in terms of Nevanlinna and d-Nevanlinna domains, that is, in terms of special analytic characteristics of simply connected domains in ℂ, which are related to the pseudocontinuation property of bounded holomorphic functions