Non-surjective Gaussian maps for singular curves on K3 surfaces

Let $(S,L)$ be a polarized K3 surface with $\mathrm{Pic}(S) = \mathbb{Z}[L]$ and $L\cdot L=2g-2$, let $C$ be a nonsingular curve of genus $g-1$ and let $f:C\to S$ be such that $f(C) \in \vert L \vert$. We prove that the Gaussian map $\Phi_{\omega_C(-T)}$ is non-surjective, where $T$ is the degree two divisor over the singular point $x$ of $f(C)$. This generalizes a result of Kemeny with an entirely different proof. It uses the very ampleness of $C$ on the blown-up surface $\widetilde S$ of $S$ at $x$ and a theorem of L'vovski.