XeLaTeX doesn't have complete fonts
The problem was that in the fonts I was trying to use this particular diacritic sing doesn't exist. Solution given to me on tex.stackexchange is to add this character in the preamble of .tex document. The code for simple XeLaTeX document containing ẖ is: 

\documentclass{article}

\usepackage{fontspec}
\usepackage{newunicodechar}

\UndeclareUTFcomposite[\UTFencname]{x1E96}{\b}{h}
\makeatletter
\DeclareRobustCommand{\b}[1]
{\hmode@bgroup\o@lign{\relax#1\crcr\hidewidth\ltx@sh@ft{-3ex}%
\vbox to.2ex{\hbox{\char"AF}\vss}\hidewidth}\egroup}
\makeatother

\newunicodechar{ẖ}{\b{h}}

\begin{document}

\b{h}ur\=u\v g

\end{document}


Other possibility is to install fonts, like Linux Libertine, which do have this character.

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