Higher-power inverse functional identities and Frobenius collision obstructions

Let $D$ be a division ring, let $n\geq 2$, and let $f,g:D\to D$ be additive maps satisfying $f(x)x^{-1}+x^n g(x^{-1})=0$ for every nonzero $x\in D$. We establish general vanishing criteria, give an explicit classification over finite fields, and determine the additive-polynomial solutions over inf