  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 {\delta}m in terms of the redshift parameter z. We observe that
{\delta}m 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 {\Lambda}CDM 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.
