We consider a single-mode laser model driven by correlated quantum and pump noise and study the effects of the cross correlation time τ between noises on the steady-state intensity correlation function and the associated relaxation time. Based on an approximated Fokker–Planck description of the laser model and the Stratonovich-like ansatz, we find the steady-state probability distribution function Pst(I), the steady-state intensity correlation function K(s) and the associated relaxation time T. From numerical computations we found the following: (1) τ suppresses the fluctuation of the laser intensity for the case of positively correlated noises (i.e. the correlation strength between noises λ > 0) and increases the intensity fluctuation for the case of negatively correlated noises (λ > 0); (2) τ slows down the decay of the intensity correlation for the case of λ > 0 but speeds up the decay for the case of λ < 0. The effects of τ on Pst(I), K(s) and T are entirely opposite for λ < 0 and for λ > 0.