Minion Pro is well known to lack lots of glyphs.
But it has U+02D8 BREVE and you can define a workaround (note: copy-paste won't work).
\documentclass{article}
\usepackage{fontspec}
\usepackage{newunicodechar}
\setmainfont{Minion Pro}
\newfontfamily{\test}{Libertinus Serif} % just to see that the correct character is produced
\NewDocumentCommand{\U}{m}{%
\iffontchar\font"032E
#1^^^^032e
\else
\makeunderbreve{#1}%
\fi
}
\newcommand{\makeunderbreve}[1]{%
\leavevmode
\vtop{\offinterlineskip
\ialign{##\hfil\cr
#1\cr
\noalign{\kern0.15ex}
\hidewidth^^^^02d8\kern10\fontdimen1\font\hidewidth\cr
\noalign{\kern-1.2ex}
}%
}%
}
\newunicodechar{ḫ}{\U{h}}
\setlength{\fboxsep}{0pt}% to show the bounding box
\begin{document}
% U+1E2B
ḫ \U{h} \fbox{\U{h}}
{\test ḫ} % this can be copy-pasted (it's in Libertinus Serif)
\textit{aḫa} \textbf{aḫa} \textbf{\textit{aḫa}}
\textit{aha} \textbf{aha} \textbf{\textit{aha}}
\end{document}

If U+032E COMBINING BREVE BELOW exists in the current font, it's used, otherwise the breve is put below with an alignment.