  Let $\Sigma$ be a closed orientable hyperbolic surface. We introduce the
notion of a \textit{geodesic current with corners} on $\Sigma$, which behaves
like a geodesic current away from certain singularities (the "corners"). We
topologize the space of all currents with corners and study its properties.
  We prove that the space of currents with corners shares many properties with
the space of geodesic currents, although crucially, there is no canonical
action of the mapping class group nor is there a continuous intersection form.
To circumvent these difficulties, we focus on those currents with corners
arising from harmonic maps of graphs into $\Sigma$. This leads to the space of
\textit{marked harmonic currents with corners}, which admits a natural Borel
action by the mapping class group, and an analog of Bonahon's\cite{Bonahon}
compactness criterion for sub-level sets of the intersection form against a
filling current.
  As an application, we consider an analog of a curve counting problem on
$\Sigma$ for triangulations. Fixing an embedding $\phi$ of a weighted graph
$\Gamma$ into $\Sigma$ whose image $\phi(\Gamma)$ is a triangulation of
$\Sigma$, let $N_{\phi}(L)$ denote the number of mapping classes $f$ so that a
weighted-length minimizing representative in the homotopy class determined by
$f \circ \phi$ has length at most $L$. In analogy with theorems of
Mirzakhani\cite{Mirzakhani}, Erlandsson-Souto\cite{ErlandssonSouto}, and
Rafi-Souto\cite{RafiSouto}, we prove that $N_{\phi}(L)$ grows polynomially of
degree $6g-6$ and the limit \[ \lim_{L \rightarrow \infty}
\frac{N_{\phi}(L)}{L^{6g-6}}\] exists and has an explicit interpretation
depending on the geometry of $\Sigma$, the vector of weights, and the
combinatorics of $\phi$ and $\Gamma$.
