The authors consider the problem of numerically computing the Hadamard finite-part integral \(\int_{-1}^1 u(t) (t-\lambda)^{-p} dt\) with \(\lambda \in (-1,1)\) and \(p \in \{1,2,3,\ldots\}\). The algorithm is based on the classical and well known principle of subtracting the singularity. This approach leads to a decomposition of the given integral into two terms, one of them can be computed analytically in closed form whereas the other one has the form of an integral with no more than a weak (improperly integrable) singularity. This latter integral is then computed numerically via a quadrature formula based on approximating the integrand with the help of a refinable quasi-interpolatory operator. From the well known convergence properties of these operators the authors then deduce sufficient conditions for the convergence of their overall scheme for the finite-part integral.