  Let $\mathcal O$ be a compact Riemannian orbisurface. We compute formulas for
the contribution of cone points of~$\mathcal O$ to the coefficient at $t^2$ of
the asymptotic expansion of the heat trace of $\mathcal O$, the contributions
at $t^0$ and $t^1$ being known from the literature. As an application, we
compute the coefficient at $t^2$ of the contribution of interior angles of the
form $\gamma=\pi/k$ in geodesic polygons in surfaces to the asymptotic
expansion of the Dirichlet heat kernel of the polygon, under a certain symmetry
assumption locally near the corresponding corner. The main novelty here is the
determination of the way in which the Laplacian of the Gauss curvature at the
corner point enters into the coefficient at $t^2$. We finish with a conjecture
concerning the analogous contribution of an arbitrary angle $\gamma$ in a
geodesic polygon.
