Upon Fourier transformation the convolution becomes a product of the Fourier transform ${\cal F}[f]$ of the function $f$ and the Fourier transformed Gaussian measure, which is again a Gaussian with covariance matrix $C^{-1}$,
$${\cal F}[\mu_{C}*f](k) = \exp\left(-\tfrac{1}{2}\sum_{n,m}k_n C_{nm} k_m\right){\cal F}[f](k).$$
Upon inverse Fourier transformation $k_n\mapsto i\partial/\partial x_n$, hence
$$(\mu_{C}*f)(x) = \exp\left(\tfrac{1}{2}\sum_{n,m}\frac{\partial}{\partial x_n} C_{nm} \frac{\partial}{\partial x_m} \right)f(x).$$
 This holds irrespective of whether $f$ is polynomial or not.