Small p with dot below

The optional argument to \stackunder specifies the gap.

\documentclass{article}
\usepackage{stackengine}
\begin{document}
ha\stackunder[-1.5pt]{p}{.}\stackunder[0pt]{p}{.}y
\end{document}

There's a COMBINING DOT BELOW (U+0323) character, but it does not look really good with EB Garamond (it works with H, but not P). However, you can adjust kerning so it fits the P letter:

%!TEX program = lualatex
\documentclass{article}
\usepackage{fontspec}
\setmainfont{EB Garamond}
\newcommand{\mydoT}[1]{#1\kern -.2em \char"0323\relax}
\begin{document}
\Huge 
p\char"0323\relax
    
p\kern -.2em \char"0323\relax

\mydoT{p}
\end{document}

Combining characters sometimes need some help if the font designer didn't set up anchors for them.

\documentclass{article}
\usepackage{fontspec}
\setmainfont{EB Garamond}

\DeclareRobustCommand{\pdb}{%
  p\llap{^^^^0323\kern0.4\fontcharwd\font`p}%
}
\DeclareRobustCommand{\gdb}{%
  g\llap{\lower0.9\fontchardp\font`g\hbox{^^^^0323\kern0.5\fontcharwd\font`g}}%
}

\begin{document}

\pdb\d{o}\gdb\d{n}

\end{document}

Here the combining character U+0323 COMBINING DOT BELOW is placed in a zero width box by itself, so it doesn't combine. A suitable kern is added at the end, which moves left the character.

The amount of adjustment has to be computed on a font basis, I'm afraid.

If you want to move down the dot below the p, use the definition of \gdb as a template.