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Sulphur is one of the most abundant elements in the Universe. Surprisingly, sulphuretted molecules are not as abundant as expected in the interstellar medium, and the identity of the main sulphur reservoir is still an open question. Our goal is to investigate the H$_{2}$S chemistry in dark clouds, as this stable molecule is a potential sulphur reservoir. Using millimeter observations of CS, SO, H$_{2}$S, and their isotopologues, we determine the physical conditions and H$_{2}$S abundances along the cores TMC 1-C, TMC 1-CP, and Barnard 1b. The gas-grain model Nautilus is then used to model the sulphur chemistry and explore the impact of photo-desorption and chemical desorption on the H$_2$S abundance. Our model shows that chemical desorption is the main source of gas-phase H$_2$S in dark cores. The measured H$_{2}$S abundance can only be fitted if we assume that the chemical desorption rate decreases by more than a factor of 10 when $n_{\rm H}>2\times10^{4}$. This change in the desorption rate is consistent with the formation of thick H$_2$O and CO ice mantles on grain surfaces. The observed SO and H$_2$S abundances are in good agreement with our predictions adopting an undepleted value of the sulphur abundance. However, the CS abundance is overestimated by a factor of $5-10$. Along the three cores, atomic S is predicted to be the main sulphur reservoir. We conclude that the gaseous H$_2$S abundance is well reproduced, assuming undepleted sulphur abundance and chemical desorption as the main source of H$_2$S. The behavior of the observed H$_{2}$S abundance suggests a changing desorption efficiency, which would probe the snowline in these cores. Our model, however, overestimates the observed gas-phase CS abundance. Given the uncertainty in the sulphur chemistry, our data are consistent with a cosmic elemental S abundance with an uncertainty of a factor of 10.