Holomorphic extension of $CR$-functions with singularities on a generic manifold

Let $\Gamma$ be a smooth generic manifold with nonzero Levi form in a domain of holomorphy $\Omega\subset\mathbf C^n$ with $n>1$. Let $\Omega_\Gamma\subset\Omega$ be the domain adjacent to $\Gamma$ to which all $CR$-functions defined on $\Gamma$ extend holomorphically. Let $K=\widehat K_\Omega\subset\Omega$ be a holomorphically convex compact set. We show that every $CR$-function on $\Gamma\setminus K$ of class $\mathscr L_{\text{loc}}^1(\Gamma\setminus K)$ extends holomorphically to $\Omega_\Gamma\setminus K$. When $n=2$ the manifold $\Gamma$ must be closed, i.e., $\partial\Gamma=0$. As a corollary we deduce a result on the removal of singularities of $CR$-functions of finite order of growth near $K$. The proof uses the integral representation of Airapetyan and Khenkin.