We study the class of Schrödinger operators whose potential terms are sums of the short-range V (r) and point potentials. We consider the case where the short-range potential has a singularity on the support r = 0 of the point interaction. The point interaction is constructed using the asymptotic form of the Green’s function of the Schrödinger operator −Δ+V (r) with a short-range potential V as r → 0 . We consider potentials with a singularity of the form r −ρ , ρ > 0 , at the origin. We use the Lippmann-Schwinger integral equation in our study. We show that if the singularity of the potential is weaker than the Coulomb singularity, then the asymptotic behavior of the Green’s function has a standard singularity. If the singularity of the potential has the form r −ρ , 1 ≤ ρ < 3/2 , then an additional singularity arises in the asymptotic behavior of the Green’s function. If ρ = 1 , then the additional logarithmic singularity has the same form as in the case of the Coulomb potential. If 1 < ρ < 3/2 , then the additional singularity has the form of the polar singularity r −ρ+1 .