Let g be a polynomial of positive degree over a finite field. Recently, Shparlinski and Weingartner gave an analogue of Romanoff’s theorem over a finite field by showing that the proportion of monic polynomials of degree n of the form \(h+g^k\) is asymptotic to \(1/\deg g\) as \(\deg g\rightarrow \infty \) and \(\deg g/n\rightarrow 0\) , where h is an irreducible monic polynomial of degree n and k is a nonnegative integer. Motivated by the result of Shparlinski and Weingartner, we prove that the proportion of monic polynomials of degree n of the form \(h+g^{k_1^2}+g^{k_2^2}\) is asymptotic to \(1/(2\deg g)\) as \(\deg g\rightarrow \infty \) and \(\deg g/n\rightarrow 0\) . Additionally, we show that the proportion of monic polynomials of degree n of the form \(h+g^{2^k}+g^{p}\) is asymptotic to \(1/(\deg g\log 2)\) under the same condition, where p is a prime.
