Let $q$ be a prime power, let $n$ be a positive integer and let $\mathbb{F}_q$ be the finite field of cardinality $q$. I have some computational evidence that the set $$\{x^n+(-1)^nay^n:x,y\in\mathbb{F}_q\}$$ is the whole of $\mathbb{F}_q$, unless $q$ is small (with respect to $n$).

This does make sense because if, for instance, $q-1=n$, then the set $\{ x^n+(-1)^nay^n:x,y\in\mathbb{F}_q \}$ only consists of $\{0,1,a,1+a\}$ because each element of $\mathbb{F}_q$ raised to the $n$ equals either $0$ or $1$.

Can anyone justify the computational evidence?