Introduced by Gutman in 2021, the Sombor index is a novel graph-theoretic topological descriptor possessing potential applications in the modeling of thermodynamic properties of compounds. Let Hk n be the family of graphs on order n and k number of cutvertices having at least one cycle. In this paper, we present minimum Sombor indices of graphs in Hk n. The corresponding extremal graphs have been characterized as well.