In this paper, we initiate the study of enriched $\rho $-nonexpansive mappings in modular function spaces. First we show that in modular function spaces, every $\rho $-nonexpansive mapping is enriched $\rho $-nonexpansive mapping but not conversely and that their sets of fixed point are same. Next, we prove a $\rho $-convergence result on approximation of fixed points of enriched $\rho $-nonexpansive mappings in modular function spaces. We verify the validity of the result by an example. We construct a table to show our findings. Finally, we give one more $\rho $-convergence result under different conditions. Our results are new for $\rho $-nonexpansive mappings in modular function spaces.