We develop the theory of Thurston maps that are defined everywhere on the topological sphere $S^2$ with a possible exception of a single essential singularity. We establish an analog of the celebrated W. Thurston's characterization theorem for a broad class of such Thurston maps having four postsingular values. To achieve this, we analyze the corresponding pullback maps defined on the one-complex dimensional Teichmüller space. This analysis also allows us to derive various properties of Hurwitz classes of the corresponding Thurston maps.