On strongly almost $m$-$\omega_1$-$p^{\omega+n}$-projective abelian $p$-groups

For any non-negative integers $m$ and $n$ we define the class of strongly almost $m$-$\omega_1$-$p^{\omega+n}$-projective groups which properly encompasses the classes of strongly $m$-$\omega_1$-$p^{\omega+n}$-projective groups and strongly almost $\omega_1$-$p^{\omega+n}$-projective groups, defined by the author in Demonstr. Math. (2014) and Hacettepe J. Math. Stat. (2015), respectively. Certain results about this new group class are proved as well as it is shown that it shares many analogous basic properties as those of the aforementioned two group classes.