The aim of the paper is to study modules for the twisted Heisenberg-Virasoro algebra at level zero as modules for the -algebra by using construction from [J. Pure Appl. Algebra 219(2015), 4322-4342, arXiv:1405.1707]. We prove that the irreducible highest weight -module is irreducible as -module if and only if it has a typical highest weight. Finally, we construct a screening operator acting on the Heisenberg-Virasoro vertex algebra whose kernel is exactly vertex algebra.
Key words: Heisenberg-Virasoro Lie algebra; vertex algebra; algebra; screening-operators.