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Let ${\mathscr M}(p)$ $(p=2,3,\ldots)$ be the singlet vertex operator algebraand $\omega$ its conformal vector. We classify the simple weak ${\mathscrM}(p)$-modules with a non-zero element $u$ such that for some integer $s\geq2$, $\omega_i u\in{\mathbb C} u$ ($i=\lfloor s/2\rfloor+1,\lfloors/2\rfloor+2,\ldots,s-1$), $\omega_{s} u\in{\mathbb C}^{\times} u$, and$\omega_i u=0$ for all $i>s$.

Simple weak modules for some subalgebras of the Heisenberg vertex algebra and Whittaker vectors