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We disclose the effects of an extra longitudinal degree of freedom on the evolution of perturbations in the framework of mimetic gravity. We consider a flat Friedmann-Robertson-Walker (FRW) background and explore the linear perturbations by adopting the spherically symmetric collapse formalism. By suitably choosing the potential of the mimetic field, we are able to solve the perturbed field equations in the linear regime and derive the matter density contrast δm in terms of the redshift parameter z. We observe that δm starts growing at the early stages and as the universe expands, it grows faster compared to the standard cosmology. This may due to the extra degree of freedom of the gravitational field which affects the growth of perturbations. We observe that in the presence of a mimetic potential, the growth rate function is smaller than ΛCDM model in small redshifts. We then consider the effects of this potential on the density abundance, the deceleration parameter and jerk parameter. We find out that mimetic potential can play the role of dark energy (DE) and affects the dynamics of matter perturbations and cosmological parameters. We also investigate the mass function and the number count for the collapsed objects in the mimetic scenario. We find that the mass function of models with potential is smaller than model without potential. With decreasing the role of DE, the mass function start to grow in smaller redshifts i.e., halo abundance is formed later. It is found that the more massive structures are less abundant and form at later times, as it should be in the hierarchical model of structure formation.