The study of the properties of the observed Higgs boson is one of the main research activities in high-energy physics. Although the couplings of the Higgs to the weak gauge bosons and third-generation quark and leptons have been studied in detail, little is known about the Higgs couplings to first- and second-generation fermions. In this article, we study the charm quark-Higgs coupling in the so-called κ framework. We emphasize the existence of specific correlations between the Higgs couplings that can render the measured LHC Higgs production rates close to the Standard Model (SM) values in the presence of large deviations of the charm coupling from its SM value, $κ_c=1$. Based on this knowledge, we update the indirect bounds on κc through a fit to the precision Higgs measurements at the LHC. We also examine the limits on $κ_c$ arising from the radiative decay $H→J/ψ+γ$, the charm-quark-associated Higgs production, charm-quark decays of the Higgs field, charge asymmetry in $W^±+H$ production, and differential production cross section distributions. Estimates for the future LHC sensitivity on κc at the high-luminosity run are provided.