Denote by \( \psi \) the finite Blaschke product, by \( H^{\infty} \) the space of bounded analytic functions on the open unit disc \( D \) and by \( \psi H^{\infty}(D)+{\mathbf C} \) - the algebra, where \( {\mathbf C} \) denotes the span of the constant functions in \( H^{| infty}(D) \). In the paper under consideration the authors describe Bourgain algebras of \( \psi H^{\infty}(D)+{\mathbf C} \), relative to \( L^{\infty}(\partial D) \) and \( C(M) \), \( M = M(H^{\infty}) \) being the spectrum of \( H^{\infty} \). More precisely, it is shown in Theorem 3.1 that the following two Bourgain algebras coincide: \( (\psi H^{\infty}(D)+{\mathbf C}, L^{\infty}(\partial D))_{b} = (H^{\infty}(\partial D), L^{\infty}(\partial D))_{b} \).