We investigate the existence and stability of discrete breathers in a chain of masses connected by linear springs and subjected to vibro-impact on-site potentials. The latter are comprised of harmonic springs and rigid constraints limiting the possible motion of the masses. Local dissipation is introduced through a non-unit restitution coefficient characterizing the impacts. The system is excited by uniform time-periodic forcing. The present work is aimed to study the existence and stability of similar breathers in the space of parameters, if additional harmonic potentials are introduced. Existence–stability patterns of the breathers in the parameter space and possible bifurcation scenarios are investigated analytically and numerically. In particular, it is shown that the addition of a harmonic on-site potential can substantially extend the stability domain, at least close to the anti-continuum limit. This result can be treated as an increase in the robustness of the breather from the perspective of possible practical applications.