This report presents a series of low-temperature (4.5 K) scanning tunneling microscopy and spectroscopy experimental results on monolayer MoS2 deposited on highly oriented pyrolytic graphite using chemical vapor deposition. To reveal the detailed connection between atomic morphology and conductivity in twisted MoS2/graphite heterojunctions, we employ high-sensitivity tunneling spectroscopy measurements by choosing a reduced tip-sample distance. We discern previously unobserved conductance peaks within the band gap range of MoS2, and by comparing the tunneling spectra from MoS2 grains of varying rotation with respect to the substrate, show that these features have small but non-negligible dependence on the moiré superstructure. Furthermore, within a single moiré supercell, atomically resolved tunneling spectroscopy measurements show that the spectra between the moiré high and low areas are also distinct. These in-gap states are shown to have an energy shift attributed to their local lattice strain, matching corresponding behavior of the conduction band edge, and we therefore infer that these features are intrinsic to the density of states, rather than experimental artifacts, and attribute them to the twisted stacking and local strain energy of the MoS2/graphite heterointerface.