Inspired by recent applications of point processes to biological nanonetworks, this paper presents a novel variant of a Matérn cluster process (MCP) in which the points located within a certain distance from the cluster centers are removed. We term this new process the MCP with holes at the cluster center (MCP-H, in short). Focusing on the three-dimensional (3D) space, we first characterize the conditional distribution of the distance between an arbitrary point of a given cluster to the origin, conditioned on the location of that cluster, for both MCP and MCP-H. These distributions are shown to admit remarkably simple closed forms in the 3D space, which is not even possible in the simpler two-dimensional (2D) case. Using these distributions, the contact distance distribution and the probability generating functional (PGFL) are characterized for both MCP and MCP-H.