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-primitive 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 [21, 57, 69], 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 [71, 72] on Gieseker moduli spaces of sheaves on K3 surfaces with non-primitive Mukai vector.