The functional quantum spin Hall insulators (QSHI), protected by time-reversal symmetry against single-particle backscattering, hold great promise for dissipationless quantum electronics. Realization of QSHI with gapped helical edge states, which would enable deterministic on/off switching of the edge-channel conductance, is a key requirement for programmable topological circuits. Here, we report realization of superlattice-modulated QSHI HfTe5 hosting emergent multigap-like helical edge states. Using scanning tunneling microscopy and spectroscopy, we identify a reconstruction-induced periodic superlattice modulation in epitaxial monolayer HfTe5 and directly observe multiple gap-like features in the edge channel, accompanied by a series of sharp peaks in the density of states. Combined with theoretical modelling, we attribute the observed edge gap to the finite-width coupling between the two edges significantly enhanced by the superlattice modulation, whereas the sharp peaks are the manifestations of mini-gaps opening at the reduced Brillouin zone boundaries by the periodic modulation of spin-orbit coupling. Notably, these sharp peaks exhibit clear Zeeman splitting under magnetic fields, consistent with the helical nature of the topological edge states. Our results establish a viable route to engineering gapped helical edge states in QSHI and provide a promising platform for topological devices with desired on/off switchability.