Primitive normal pairs of elements with one prescribed trace

Let $q,n,m\in \mathbb{N}$ such that $q$ is a prime power, $m\ge 3$ and $a\in \mathbb{F}$. We establish a sufficient condition for the existence of a primitive normal pair ($\alpha$, $f(\alpha )$) in ${\mathbb{F}}_{q^m}$ over ${\mathbb{F}}_q$ such that Tr${}_{{\mathbb{F}}_{q^m}\mathrm{/}{\mathbb{F}}_q}({\alpha }^{-1})=a$, where $f(x)\in {\mathbb{F}}_{q^m}(x)$ is a rational function with degree sum $n$. In particular, for $q=5^k,k\ge 5$ and degree sum $n=4$, we explicitly find at most 11 choices of $(q,m)$ where existence of such pairs is not guaranteed.