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The formation of clusters at sub-saturation densities constitutes an essential feature for a reliable modelization of the nuclear matter equation of state (EoS). Phenomenological models that make use of energy density functionals (EDFs) offer a convenient approach to account for the presence of these bound states of nucleons when introduced as additional degrees of freedom. However, in these models clusters dissolve, by construction, when the nuclear saturation density is approached from below, revealing inconsistencies with recent findings that evidence the existence of short-range correlations (SRCs) even at larger densities. In this work, within the EDF framework, a novel approach is proposed to embed SRCs within a relativistic mean-field model with density dependent couplings. This is realized through the introduction of suitable in-medium modifications of the cluster binding energy shifts, which are responsible for describing the cluster dissolution. As a first exploratory step, the example of a quasi-deuteron within the generalized relativistic density functional approach is investigated. For the first time, suitable parameterizations of the cluster mass shift at zero temperature are derived for all baryon densities. They are constrained by experimental results for the effective deuteron fraction in nuclear matter near saturation and by microscopic many-body calculations in the low-density limit. The strength of the deuteron-meson couplings is assessed to be of crucial importance. The findings of the present study represent a first step to improve the description of nuclear matter and its EoS at supra-saturation densities in EDFs by considering correlations in an effective way. Novel effects on some thermodynamic quantities, such as the matter incompressibility, the symmetry energy and its slope, are finally discerned and discussed., 28 pages, 12 figures