  We prove an enhanced limit theorem for additive functionals of a
multi-dimensional Volterra process $(y_t)_{t\geq 0}$ in the rough path
topology. As an application, we establish weak convergence as $\varepsilon\to
0$ of the solution of the random ordinary differential equation (ODE)
$\frac{d}{dt}x^\varepsilon_t=\frac{1}{\sqrt \varepsilon}
f(x_t^\varepsilon,y_{\frac{t}{\varepsilon}})$ and show that its limit solves a
rough differential equation driven by a Gaussian field with a drift coming from
the L\'evy area correction of the limiting rough driver. Furthermore, we prove
that the stochastic flows of the random ODE converge to those of the Kunita
type It\^o SDE $dx_t=G(x_t,dt)$, where $G(x,t)$ is a semi-martingale with
spatial parameters.
