An explicit polynomial analogue of Romanoff's theorem

Given a polynomial $g$ of positive degree over a finite field, we show that the proportion of polynomials of degree $n$, which can be written as $h+g^k$, where $h$ is an irreducible polynomial of degree $n$ and $k$ is a nonnegative integer, has order of magnitude $1\mathrm{/}\mathrm{deg}g$.