Existence of primitive normal pairs with one prescribed trace over finite fields

Given $$m, n, q\in \mathbb {N}$$ such that q is a prime power and $$m\ge 3$$ , $$a\in {F}_q$$ we establish sufficient condition for the existence of primitive pair $$(\alpha f(\alpha ))$$ in $$\mathbb {F}_{q^m}$$ $$\alpha $$ normal over $$\text {Tr}_{\mathbb {F}_{q^m}/\mathbb {F}_q}(\alpha ^{-1})=a$$ where $$f(x)\in {F}_{q^m}(x)$$ rational function degree sum n. Further, when $$n=2$$ $$q=5^k$$ some $$k\in definitely exists all (q, m) apart from at most 20 choices.