We study the edge-averaging process on a finite, connected graph $G = (V, E)$. Initially, the vertices in $V$ are endowed with i.i.d.\ real-valued opinions $(f_0(v))_{v \in V}$. Edges are activated according to i.i.d.\ Poisson clocks of rate $1$; when an edge is activated, the opinions at its endpoints are replaced by their average. Let $f_t(v)$ denote the opinion at $v$ at time $t$.Define the $ε$-convergence time $τ_ε$ as the first time when the maximum and the minimum of $f_t$ differ by at most $ε$. It is known that if the initial opinions $(f_0(v))_{v \in V}$ are bounded in $L^\infty$, then $\mathbb{E}(τ_ε)$ is at most $C_ε\log^2 n$ for $ε\in (0, 1]$. We assume instead that the $L^p$ norm of $f_0(v)$ is at most $1$ for every $v \in V$. For fixed $ε\in (0, 1]$, and show that $\mathbb{E}(τ_ε) = \widetilde{O}(n^{β_p})$ up to logarithmic terms, where $β_p := \max(3 - p, 2/p)$. Moreover, this power law is tight on cycle graphs.