We report an experiment of the time-delayed Raman-enhanced nondegenerate four-wave mixing (RENFWM) with a broadband laser in carbon disulfide (${\mathrm{CS}}_{2}$). We studied the RENFWM signal intensity as a function of the relative time delay \ensuremath{\tau} between two beams which originate from a single broadband laser source. Our results indicate that the temporal behavior of the RENFWM is asymmetric with the maximum of the signal shifted from \ensuremath{\tau}=0. Furthermore, unlike the corresponding coherent Stokes Raman scattering (CSRS) no coherent spike appears at \ensuremath{\tau}=0. From our experimental results the relaxation time of the Raman mode can be extracted directly. We present a second-order coherence function theory to elucidate the physics of the basic features of the time-delayed RENFWM. We also discuss the difference between the RENFWM and the CSRS from a physical viewpoint.