  The Hanna Neumann Conjecture (HNC) for a free group $G$ predicts that
$\overline\chi(U \cap V) \leqslant \overline\chi(U) \overline\chi(V)$ for all
finitely generated subgroups $U$ and $V$, where $\overline\chi(H) =
\min\{-\chi(H),0\}$ denotes the reduced Euler characteristic of $H$. A
strengthened version of the HNC was proved independently by Friedman and
Mineyev in 2011. Recently, Antol\'in and Jaikin-Zapirain introduced the
$L^2$-Hall property and showed that if $G$ is a hyperbolic limit group that
satisfies this property, then $G$ satisfies the HNC. Antol\'in--Jaikin-Zapirain
established the $L^2$-Hall property for free and surface groups, which
Brown--Kharlampovich extended to all limit groups. In this article, we prove
the $L^2$-Hall property for graphs of free groups with cyclic edge groups that
are hyperbolic relative to virtually abelian subgroups and also give another
proof of the $L^2$-Hall property for limit groups. As a corollary, we show that
all these groups satisfy a strengthened version of the HNC.
