Single-photon emission centers generated by controlled atomic force microscopy (AFM) indentation in monolayer WSe\(_2\) on a flexible polymer substrate are explored for applications in quantum technologies. Here, we study the response of these emitters to the polymer substrate's strain state, which is controlled by selecting the indentation force and by gradually thermally annealing the samples. In the indented areas, we observe sharp new photoluminescence (PL) peaks in the regions 1.62--1.68 eV and 1.70--1.73 eV, characterized by sublinear power dependence and spectral wandering. We find that these peaks arise only when the indentation force exceeds a few \(μ\)N and generally redshift as the applied force increases. Conversely, after thermal annealing (\(T < 60^{\circ}\)C), WSe\(_2\) experiences strain relaxation, leading to a blueshift of the peaks' spectral position and their ultimate disappearance. Our analysis of the peaks' positions vs. strain allows us to draw several conclusions about the nature of these emission. Specifically, we elucidate the roles of excitonic confinement and hybridization between free excitons and defect-related states, a process activated by the strain level. Overall, our approach suggests that the energy of localized emitters may be controlled via strain engineering.