Given a projective variety $X$ over a field of any characteristic, consider a line bundle $\mathcal{L}$ over $X$.

The existence of a line bundle $\mathcal{L}^\prime$ with an isomorphism ${\mathcal{L}^\prime}^{\otimes 2} \simeq \mathcal{L}$ is equivalent to the existence of a simple cyclic cover $Y \rightarrow X$ of degree $2$ which is branched on a divisor $D$ associated to $\mathcal{L}$. Such a cyclic cover depends on the choice of the rational section $s$ of $\mathcal{L}$ such that $D = \text{div}(s)$. See section 3.3 of https://arxiv.org/abs/2009.01831v2 for a proof. In the related question divisors and powers of line bundles, Francesco Polizzi gives an example of a line bundle on a K3 surface which does not admit a square root.

There is also the root stack construction that gives a Deligne-Mumford stack $f : {}^{2}\sqrt{(X,\mathcal{L})} \rightarrow X$ which is the moduli space of square roots of $\mathcal{L}$.

I am wondering if there exists a finite surjective morphism of projective varieties $g : X^\prime \rightarrow X$ such that $g^*\mathcal{L}$ admits a square root over $X^\prime$. If such a map exists, it should factor through $f$.