  This article studies moduli spaces of Bridgeland semistable objects in the
Kuznetsov component of a cubic fourfold that don't admit a symplectic
resolution, i.e., moduli spaces of objects with non-primitve Mukai vector
v=mv_0 that is not of OG10-type and where v_0^2 >0. For a generic stability
condition, it is shown that these moduli spaces are projective irreducible
symplectic varieties with factorial terminal singularities and that their
deformation class is uniquely determined by the integers m and v_0^2. On the
one hand, this generalizes the results of arXiv:1703.10839, arXiv:1912.06935,
arXiv:2007.14108, which deal with moduli spaces of objects in the Kuznetsov
component of a cubic fourfold which are smooth or of OG10-type; on the other
hand, this extends to the Kuznetsov component of a cubic fourfold the results
of arXiv:1802.01182, arXiv:2012.10649, on Gieseker moduli spaces of sheaves on
K3 surfaces with non-primitive Mukai vector.
