Recently, Nguyen, Seymour and Scott verified the small quasi-kernel conjecture for split digraphs, and initiated the study of quasi-kernels in break digraphs. Following their research, we introduce a weighted half-neighborhood property for hereditary classes of oriented graphs and show that it gives a \(2n/3\) bound of small quasi-kernel for break digraphs. We also record two stronger \(n/2\) results for special classes of break digraphs. Finally, using the same framework we also prove that every digraph on \(n\) vertices has a quasi-kernel \(Q\) with \(|N_D^+[Q]|\ge \sqrt n\).