Regularization of a distribution holomorphic in a parameter

We give sufficient conditions for regularizing a distribution of the form $a(\sigma,\lambda)f(\lambda)$, where $f(\lambda)$ is a distribution holomorphic in the parameter $\lambda$, while $a(\sigma, \lambda)$ is an infinitely differentiable function of $\sigma$ outside some closed set $N$ with power singularities of derivatives on $N$ and holomorphic in $\lambda$.