A mixed cavity optomechanical system consists of both the first-order and quadratic optomechanical interactions. We study the two-photon scattering problem in the mixed cavity optomechanical system. By solving the scattering process within the Wigner-Weisskopf framework, we obtain the analytical expression of the scattering state and find four physical processes associated with the two-photon scattering process in this system: 1) two photons are directly reflected by the fixed end mirror, without entering the cavity; 2) one photon is reflected directly and the other photon enters the cavity; 3) two photons enter the cavity in sequence, but there is at most one photon in the cavity; 4) two photons enter the cavity. By analyzing the two-photon scattering spectrum, we find that the two-photon frequency anticorrelation can be induced via this scattering process. We also find the relationship between the parameters of the mixed cavity optomechanical system and the characteristics of the two-photon scattering spectrum. This study not only provides a scattering method to create correlated-photon pairs, but also presents a spectrometric way to characterize the optomechanical system.