Let $\Omega$ be an open subset of $\mathbb{R}^3$, and let $E$ be a $L^1_{\mathrm{loc}}$ vector field on $\Omega$. We say that $E$ is curl-free in the weak sense if $E$ is a weak solution to $\nabla \times E = 0$. In other words, for any scalar test function $\psi \in C_c(\Omega)$, we have
$$ \int_\Omega E \times \nabla \psi = 0 $$
Let $p \in \Omega$. Does there exist some neighborhood $U$ of $p$ and some $L^1_{\mathrm{loc}}$ scalar function $f: U \rightarrow \mathbb{R}$ such that $\nabla f = E$ in the weak sense?
Recall that $\nabla f = E$ in the weak sense if and only if for any test function $\psi \in C_c(U)$, we have
$$ \int_U E \cdot \nabla \psi = \int_U f \psi $$
I've been able to prove this when $E$ is an $L^p_{\mathrm{loc}}$ vector field, $p > 3$, but nothing with less regularity. In case it is useful, I'll put a sketch of that proof below.
For simplicity, let $\Omega$ be a ball of radius 1 and center 0. If $E$ is smooth curl-free vector field on $\Omega$, then we can define $f(x)$ as the line integral of $E$ over any line $h(t)$ connecting $0$ to $x$. This is well-defined because $E$ is curl-free and $\Omega$ is simply connected. Furthermore, $\nabla f = E$. We can bound $|f(x) - f(0)|$ by parameterizing this line $h(t)$ by two variables $(\alpha, \beta)$ where $0 \leq \alpha \leq 1$ and $0 \leq \beta \leq 1$ and use the integral
$$ f(x) - f(0) = \int E(h(t, \alpha, \beta)) \cdot \frac{\partial h}{\partial t}(t, \alpha, \beta) \mathrm{d}t \mathrm{d}\alpha \mathrm{d}\beta $$
This integral is bounded by some multiple of the $L^p$ norm of $E$. The details get a bit hairy, but the important thing is that $|f| \ll \|E\|_{L^p}$.
Now given an arbitrary $L^p$ vector field $E$, approximate $E$ by smooth vector fields $E_m = E * \eta_m$ where $\eta_m$ is a smooth mollifier (i.e. ``approximate identity''). Let $f_m$ be the solution to $\nabla f_m = E_m$ defined above. Because $E_m$ converges in the $L^p$ norm, $f_m$ will converge uniformly. The limit of $f_m$ is some continuous function $f$ such that $\nabla f = E$, in the weak sense.