In recent years, various novel concepts of efficient low-frequency acoustic absorbers have been proposed. The present paper investigates in detail the non-linear acoustic response of resonators coupled to plates with thin slits. A laser-cut thin plate is positioned inside the neck of a Helmholtz resonator. Various shapes of slits are considered, including a cantilever beam. This system produces a strong acoustic response in the vicinity of the resonance frequency of the beam, which can be adjusted to target low frequencies. However, this acoustic response exhibits a significant dependence on the incident sound level. In this paper, these systems are mechanically and acoustically modelled and characterised under normal incidence in order to describe and predict the non-linearities observed. To this end, a generalised expression for the resistance and reactance of such thin structures is proposed and validated against experimental data.