By lemma 4.9 in Dugger-Hollander-Isaksen, a hypercover is defined as an augmented simplicial object $U_\bullet\to X$ in the category of simplicial presheaves such that each $U_n$ is a coproduct of representables and each $U_n\to (cosk_{n-1} U_\bullet)_n$ is a local epimorphism.

In the answer of this post, it shows the relation of hypercover for simplicial presheaves with hypercovering in $\infty$-topos. But the definition in the question doesn't require $U_n$ to be a coproduct of representables. Is the condition that $U_n$ being a coproduct of representables not essential for the definition of hypercover of simplicial presheaves?