  Few-photon optomechanical effects are not only important physical evidences
for understanding the radiation-pressure interaction between photons and
mechanical oscillation, but also have wide potential applications in modern
quantum technology. Here we study the few-photon optomechanical effects
including photon blockade and generation of the Schr\"{o}dinger cat states
under the assistance of a cross-Kerr interaction, which is an inherent
interaction accompanied the optomechanical coupling in a generalized
optomechanical system. By exactly diagonalizing the generalized optomechanical
Hamiltonian and calculating its unitary evolution operator, we find the
physical mechanism of the enhancement of photon blockade and single-photon
mechanical displacement. The quantum properties in this generalized
optomechanical system are studied by investigating the second-order correlation
function of the cavity field and calculating the Wigner function and the
probability distribution of the rotated quadrature operator for the mechanical
mode. We also study the influence of the dissipations on the few-photon
optomechanical effects.
