Summary: In this paper, we consider one-dimensional Schrödinger operators \(S_q\) on \(\mathbb{R}\) with a bounded potential \(q\) supported on the segment \([h_0,h_1]\) and a singular potential supported at the ends \(h_0, h_1\). We consider an extension of the operator \(S_q\) in \(L^2(\mathbb{R})\) defined by the Schrödinger operator \(\mathcal{H}_q=-\frac{d^2}{dx^2}+q\) and matrix point conditions at the ends \(h_0, h_1\). By using the spectral parameter power series method, we derive the characteristic equation for calculating the discrete spectra of operator \(\mathcal{H}_q\). Moreover, we provide closed-form expressions for the eigenfunctions and associate functions in the Jordan chain given in the form of power series of the spectral parameter. The validity of our approach is proven in several numerical examples including self-adjoint and nonself-adjoint problems involving general point interactions described in terms of \(\delta\)- and \(\delta^\prime\)-distributions.