We prove anomalous-diffusion scaling for a one-dimensional stochastic kinetic dynamics, in which the stochastic drift is driven by an exogenous Bessel noise, and also includes endogenous volatility which is permitted to have arbitrary dependence with the exogenous noise. We identify the superdiffusive scaling exponent for the model, and prove a weak convergence result on the corresponding scale. We show how our result extends to admit, as exogenous noise processes, not only Bessel processes but more general processes satisfying certain asymptotic conditions.