Let $u:\Sigma\to X$ be a holomorphic disc which is immersed near some point $z_{0}\in\partial\Sigma$ with the boundary injectivity property $u|_{\partial\Sigma}^{-1}(u(z_{0}))=\{z_{0}\}$. Let $\eta$ be a nonzero dualized cokernel element for the ordinary linearized Cauchy-Riemann operator. Then there is a Hamiltonian $H$ supported in any prescribed small ball on $X$ containing $u(z_{0})$, such that $\int_{\Sigma}\langle\bar{\partial}X_{H}\wedge\eta\rangle\neq 0$.