We define and investigate the class of almost $\omega_1$-$n$-simply presented $p$-torsion abelian groups, which class properly contains the subclasses of almost $n$-simply presented groups and $\omega_1$-$n$-simply presented groups, respectively. The obtained results generalize those obtained by us in Korean J. Math. (2014) and J. Algebra Appl. (2015).