On the almost palindromic width of certain free constructions of groups

We provide a general structural criterion implying that a group has infinite $m$-almost palindromic width. In particular, we prove that both HNN extensions and free products exhibit infinite $m$-almost palindromic width, with the unique exception of the infinite dihedral group among free products. This framework extends and strengthens the results of \cite{MS} and \cite{GK}.