  A new metric parameter for a graph, Helly-gap, is introduced. A graph $G$ is
called $\alpha$-weakly-Helly if any system of pairwise intersecting disks in
$G$ has a nonempty common intersection when the radius of each disk is
increased by an additive value $\alpha$. The minimum $\alpha$ for which a graph
$G$ is $\alpha$-weakly-Helly is called the Helly-gap of $G$ and denoted by
$\alpha(G)$. The Helly-gap of a graph $G$ is characterized by distances in the
injective hull $\mathcal{H}(G)$, which is a (unique) minimal Helly graph which
contains $G$ as an isometric subgraph. This characterization is used as a tool
to generalize many eccentricity related results known for Helly graphs
($\alpha(G)=0$), as well as for chordal graphs ($\alpha(G)\le 1$),
distance-hereditary graphs ($\alpha(G)\le 1$) and $\delta$-hyperbolic graphs
($\alpha(G)\le 2\delta$), to all graphs, parameterized by their Helly-gap
$\alpha(G)$. Several additional graph classes are shown to have a bounded
Helly-gap, including AT-free graphs and graphs with bounded tree-length,
bounded chordality or bounded $\alpha_i$-metric.
