  The presence of chalcogen vacancies in monolayer transition metal
dichalcogenides (TMDs) leads to excitons with mixed localized-delocalized
character and to reduced valley selectivity. Recent experimental advances in
defect design in TMDs allow for a close examination of such mixed exciton
states as a function of their degree of circular polarization under external
magnetic fields, revealing strongly varying defect-induced magnetic properties.
A theoretical understanding of these observations and their physical origins
demands a predictive, structure-sensitive theory. In this work, we study the
effect of chalcogen vacancies on the exciton magnetic properties in monolayer
MoS$_2$. Using many-body perturbation theory, we show how the complex excitonic
picture associated with the presence of defects -- with reduced valley and spin
selectivity due to hybridized electron-hole transitions -- leads to
structurally-controllable exciton magnetic response. We find a variety of
g-factors with changing magnitudes and sign depending on the exciton energy and
character. Our findings suggest a pathway to tune the nature of the excitons --
and by that their magneto-optical properties -- through defect architecture.
