We study the classical problem of identifying the structure of P2(μ), the closure of analytic polynomials in the Lebesgue space L2(μ) of a compactly supported Borel measure μ living in the complex plane. In his influential work, Thomson [Ann. of Math. (2) 133 (1991), pp. 477-507] showed that the space decomposes into a full L2-space and other pieces which are essentially spaces of analytic functions on domains in the plane. For a family of measures μ supported on the closed unit disk D which have a part on the open disk D which is similar to the Lebesgue area measure, and a part on the unit circle T which is the restriction of the Lebesgue linear measure to a general measurable subset E of T, we extend the ideas of Khrushchev and calculate the exact form of the Thomson decomposition of the space P2(μ). It turns out that the space splits according to a certain natural decomposition of measurable subsets of T which we introduce. We highlight applications to the theory of the Cauchy integral operator and de Branges-Rovnyak spaces.