Let $R$ be the finite field with $q$ elements, and let $m,n\in \mathbb{N}$
be positive integers $\geq 2$. I want to prove that there exists a primitive 
polynomial $$F(x) = x^{mn}-\sum\limits\_{j=0}^{mn-1}f\_jx^j\in R[x]$$
with the property that there is a $k\in\{1,2,\ldots,n-1\}$ such that $f\_{km}\neq 0$.
I think that this is always true except in case $q=m=n=2$.



Non-zero coefficients of primitive polynomials
I post here a Hansen-Mullen's conjecture, which was proved in 2007. The validity of my assumption in the question is a straightforward consequence of this conjecture.


**Conjecture**.(Hansen and Mullen, 1992)
*Let $a\in \mathbb{F}\_q$ and let $n\geq2$ be a positive integer. Fix and integer $m$ with $0<m<n$. Then there exists a primitive polynomial $f(x) = x^n
+\sum\_{j=1}^{n}a\_jx^{n-j}$ over $\mathbb{F}\_q$ with $a\_m = a$ with (genuine) exceptions when*
$$
(q,n,m,a) = (q,2,1,0), (4,3,1,0), (4,3,2,0), (2,4,2,1).
$$
The proof is carried out in several stages. See for details Stephen D Cohen, [Primitive polynomials with a prescribed coefficient](http://www.sciencedirect.com/science/article/pii/S1071579705000663), Finite Fields and Their Applications 12 (2006) 425-491. The completion of the proof is given in Cohen and Presern, The Hansen-Mullen primitivity conjecture: completion of proof, in McKee and Smyth, eds., [Number Theory and Polynomials](http://www.cambridge.org/ru/academic/subjects/mathematics/number-theory/number-theory-and-polynomials), London Mathematical Society Lecture Note Series 352.


