  We complete the classification of the smooth, closed, oriented 4-manifolds
having Euler characteristic less than four and a horizontal handlebody
decomposition of genus one. We use the classification result to find a large
family of rational homology ball smoothings of cyclic quotient singularities
which can be smoothly embedded into the complex projective plane. Our family
contains all such rational balls previously known to embed into $\mathbb{CP}^2$
and infinitely many more. We also show that a rational ball of our family
admits an almost-complex embedding in $\mathbb{CP}^2$ if and only if it admits
a symplectic embedding.
