Text below equation with hf-tikz

I'd use an array with top alignment:

\documentclass[aspectratio=169]{beamer}

\usepackage{tikz}
\usetikzlibrary{calc}
\usepackage{mathtools}
\usepackage{hf-tikz}

\begin{document}

\begin{frame}
\frametitle{Text under equation blocks}

\[
a = 
\onslide<2->%
  \begin{array}[t]{@{}c@{}}
  \tikzmarkin{a}\onslide<1-> b\onslide<2->\tikzmarkend{a}\\
  \scriptstyle\text{\alert{text1}}
  \end{array}
\onslide<1->+ 
\onslide<3->%
  \begin{array}[t]{@{}c@{}}
  \tikzmarkin{b}\onslide<1-> c\onslide<3->\tikzmarkend{b}\\
  \scriptstyle\text{\color{brown}text2}
  \end{array}
\]

\end{frame}

\end{document}

_{The slight misalignment is due to manually taking screen shots}

Using the markings option of the package one might derive an \annotate command:

\newcommand{\annotate}[2][]{
\tikz[remember picture,overlay]\node[#1,use marker id] at (0,0){#2};
}

A complete example:

\documentclass[aspectratio=169]{beamer}

\usepackage{lmodern} 
\usepackage{mathtools}
\usepackage[beamer,markings]{hf-tikz}% hf-tikz load itself tikz and the calc library
\newcommand{\annotate}[2][]{
\tikz[remember picture,overlay]\node[#1,use marker id] at (0,0){#2};
}

  \tikzset{
    invisible/.style={opacity=0},
    visible on/.style={alt={#1{}{invisible}}},
    alt/.code args={<#1>#2#3}{%
      \alt<#1>{\pgfkeysalso{#2}}{\pgfkeysalso{#3}} % \pgfkeysalso doesn't change the path
    },
  }
\begin{document}

\begin{frame}
\frametitle{Text under equation blocks}

\begin{equation*}
\begin{aligned}
a = \phantom{aaaaa}\tikzmarkin<1->[mark at=0.825]{a}b\tikzmarkend{a}\annotate[visible on=<1->,below,red,font=\scriptsize]{Model Mismatch}
\phantom{aaaaa} +\phantom{Noi} \tikzmarkin<2->[mark at=0.825]{b}c\tikzmarkend{b}\annotate[visible on=<2->,below,brown,font=\scriptsize]{Noise}
\end{aligned}
\end{equation*}


\end{frame}

\end{document}

The key points are:

1. mark the positions with the mark at key (to verify where the location is marked you might use in combination the show markers key);
2. insert some phantom space between and after the variable highlighted as the annotation text is too much larger with respect to the variables.

The result:

Improved version

According to egreg's comment, here is a more automatic solution exploiting \mathmakebox from the mathtools package. It provides a new \tikzmarkaligned command which sets the space according to the annotation width:

\documentclass[aspectratio=169]{beamer}

\usepackage{lmodern} 
\usepackage{mathtools,xparse}
\usepackage[beamer,markings]{hf-tikz}% hf-tikz load itself tikz and the calc library
\newcommand<>{\annotate}[2][]{
 \onslide#3{
  \tikz[remember picture,overlay]\node[#1,use marker id] at (0,0){#2};
 }
}

\newlength{\notewidth}

\newcommand{\setnotewidth}[1]{%
  \settowidth{\notewidth}{$#1$}%
}

\newcommand{\mth}[2]{
\setnotewidth{#2}
\mathmakebox[0.355\notewidth]{#1}
}

\NewDocumentCommand{\tikzmarkaligned}{r<> o m m}{
\phantom{\mth{#3}{#4}}
\tikzmarkin<#1>[#2]{#3}#3\tikzmarkend{#3}
\phantom{\mth{#3}{#4}}
}

\begin{document}

\begin{frame}
\frametitle{Text under equation blocks}

\begin{equation*}
\begin{aligned}
a = 
\tikzmarkaligned<1->[mark at=0.825]{a}{Model Mismathc}
\annotate<1->[below,red,font=\scriptsize]{Model Mismatch}+
\tikzmarkaligned<2->[mark at=0.825]{b}{Noise}
%\tikzmarkin<2->[mark at=0.825]{b}c\tikzmarkend{b}
\annotate<2->[below,brown,font=\scriptsize]{Noise}
\end{aligned}
\end{equation*}

\end{frame}