I am trying to find (as many as possible) elements in the algebraic closure of a positive characteristic field, being roots of irreducible polynomial inside some splitting field which are not roots of unity (which are not $k$-th roots of unity for any $k$). By "as many as possible" I mean I am looking for positive results on the existence of irreducible polynomials having such roots in a splitting field, like for instance ensuring one in each degree...
But proving there is one such element would already be very good for me.
No.
Every element $\theta$ of the algebraic closure of a finite field $F$ is algebraic over $F$ and so has finite degree over $F$ and thus lies in a finite extension of $F$. This extension is a finite field and so $\theta$ is a root of unity.