We report on scanning tunneling microscopy and spectroscopy studies of the slightly underdoped cuprate superconductor ${\mathrm{HgBa}}_{2}{\mathrm{Ca}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{8+\ensuremath{\delta}}$, which has an onset ${T}_{c}\ensuremath{\approx}120$ K. Disordered HgO-terminated surfaces were observed after sample cleaving in ultrahigh vacuum, revealing spatially inhomogeneous energy gaps with a low-lying $\mathsf{V}$-shaped structure and the ubiquitous mode features. Similar to Bi-family cuprate superconductors, vortex cores were identified by the suppression of superconducting coherence peaks and the emergence of in-gap features. Superconductivity near step edges is also strongly suppressed over several coherence lengths, forming an effective interface to the superconducting region, where the local spectra show a weak zero-bias conductance peak arising from the Andreev bound states. These results are expected from the widely believed $d$-wave nature in cuprates, demonstrating a platform for studying unconventional superconductivity.