Romanoff type theorems for polynomials over finite fields

Given a polynomial \(g\) of degree \(δ>0\) over a finite field, we study monic polynomials \(f\) of degree \(n\) that can be written as \(f=h+g^k\), where \(h\) is a monic irreducible polynomial of degree \(n\) and \(k\in\mathbb{N}\) with \(δk<n\). Following Erdős, we show that some \(f\) admit at l