  By definition primitive and $2$-primitive elements of a finite field
extension $\mathbb{F}_{q^n}$ have order $q^n-1$ and $(q^n-1)/2$, respectively.
We have already shown that, with minor reservations, there exists a primitive
element and a $2$-primitive element $\xi \in \mathbb{F}_{q^n}$ with prescribed
trace in the ground field $\mathbb{F}_q$. Here we amend our previous proofs of
these results, firstly, by a reduction of these problems to extensions of prime
degree $n$ and, secondly, by deriving an exact expression for the number of
squares in $\mathbb{F}_{q^n}$ whose trace has prescribed value in
$\mathbb{F}_q$. The latter corrects an error in the proof in the case of
$2$-primitive elements. We also streamline the necessary computations.
