The bound $C(1) \geq \frac{1}{\sqrt{2 \pi}}$ follows from "Bobkov's Inequality" [Bob97], which says that for any $f : \{-1,1\}^n \to [0,1]$ it holds that $\mathcal{U}(\mathbb{E}[f]) \leq \mathbb{E}\sqrt{\mathcal{U}(f)^2 + |\nabla f|^2}$, where $\nabla f$ is the discrete gradient $\nabla f(x) = \left(\frac{f(x_1, \dots, x_{i-1}, 1, x_{i+1}, \dots, x_n) - f(x_1, \dots, x_{i-1}, -1, x_{i+1}, \dots, x_n)}{2}\right)_{i=1\dots n}$ and $\mathcal{U}$ denotes the "Gaussian isoperimetric function" $\phi \circ \overline{\Phi}^{-1}$ (with $\phi$ the Gaussian pdf and $\overline{\Phi}$ the Gaussian complementary cdf). In particular, if $f$ is the $0$-$1$ indicator of a set $A \subseteq \{-1,1\}^n$, then this reduces to $\mathbb{E}_x[\sqrt{w_A(x)}] \geq \mathcal{U}(\text{vol}(A))$, where $\text{vol}(A) = |A|/2^{n}$. Thus in your case where $\text{vol}(A) = 1/2$ we get a lower bound of $\mathcal{U}(1/2) = \frac{1}{\sqrt{2 \pi}}$.

This is sharp up to a factor of $\sqrt{2}$ by virtue of the "Majority" function -- i.e., $A$ being a Hamming ball of radius $n/2$ -- which has $\mathbb{E}_x[\sqrt{w_A(x)}] \sim \frac{1}{\sqrt{\pi}}$. Bobkov remarks on this missing factor of $\sqrt{2}$; presumably $\frac{1}{\sqrt{\pi}}$ is the correct answer but it's not known how to improve $\frac{1}{\sqrt{2\pi}}$ as far as I'm aware.

Given this result, one can get $C(p) \geq \left(\frac{1}{2\pi}\right)^{p/2}$ for any $1 < p < 2$ by monotonicity of $\ell_p$-norms, which is a decent bound but is surely not optimal. Getting the optimal answer in this range is presumably harder than getting the optimal answer for $p = 1$.

By the way, it was Talagrand who first studied the $p = 1$ case; this notion of looking at the average square-root of the number of sensitive coordinates seems to be a pretty good notion of "discrete surface area". Talagrand's original paper [Tal93] proved Bobkov's Inequality (for sets $A$) up to a universal constant factor on the LHS; in particular, he established $C(1)$ is bounded away from $0$ by a univeral constant.

[Bob97] *Bobkov, S.G.*, **An isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in Gauss space**, Ann. Probab. 25, No.1, 206-214 (1997). ZBL0883.60031.

[Tal93] *Talagrand, M.*, **Isoperimetry, logarithmic Sobolev inequalities on the discrete cube, and Margulis' graph connectivity theorem**, Geom. Funct. Anal. 3, No.3, 295-314 (1993). ZBL0806.46035.