Why does foreach loop seem to interfere with let code????? See the code below. Outside a loop, repeating the same code twice works fine

Use globallet instead of let inside the foreach loop. The loop is group limited, and so using just let makes a local change, which is lost at the end of each iteration.

documentclass[ aspectratio = 1610 ]{beamer} 
 
 usepackage{psfrag}
 
 usepackage{sidecap}
 
 usepackage{tikz}
 
 usepackage{pgfplots}
 
 usepackage{amsmath}
 
  begin{document}
  
 begin{frame}
 
 begin{tikzpicture} [scale = 0.6]
 
 defscore{10}
 
  draw (0,3) node [right] {Score is initialized to 10 and will be augmented  by 3 twice by repeating code};

  pgfmathparse{score + 3 } letscorepgfmathresult

 draw  (0,2)  node [right] { New score quad    $score$ correct};

  pgfmathparse{score + 3 } letscorepgfmathresult

 draw  (0,1)  node [right] {New score quad    $score$ correct };

 draw [dashed](0, 0.5)--(20,.5);

 draw(0,0) node[right] {Score re-initialized   to 10  and will be augmented  by 3 in a  foreach loop , again twice};

 defscore{10}

  foreach k in { 1,2 }  {

  pgfmathparse{score + 3 } globalletscorepgfmathresult

  draw  (0,   -  k)  node [right] {    New score    $score$    ifthenelse{k=1} {  Correct} {should be 16  Wrong.  Why?}};
 }
 end{tikzpicture}
 
 end{frame}
  
 end{document}

Assignments in foreach loops are local to the current loop cycle. You can use the index k.

Are you really wanting to load psfrag?

documentclass[ aspectratio = 1610 ]{beamer} 
% usepackage{psfrag} % really?
%usepackage{sidecap}
usepackage{tikz}
usepackage{pgfplots}
%usepackage{amsmath}% already loaded by beamer
 
begin{document}
  
begin{frame}
 
begin{tikzpicture} [scale = 0.6]
defscore{10}
draw (0,3) node [right] {Score is initialized to 10 and will be 
                          augmented  by 3 twice by repeating code};
pgfmathsetmacroscore{score + 3 }
draw  (0,2)  node [right] { New score quad    $score$ correct};
pgfmathsetmacroscore{score + 3 }
draw  (0,1)  node [right] {New score quad    $score$ correct };
draw [dashed](0, 0.5)--(20,.5);
draw(0,0) node[right] {Score re-initialized to 10 and will be
                        augmented by 3 in a foreach loop, again twice};
defscore{10}
foreach k in { 1,2 } {
  pgfmathsetmacroscore{score + 3*k}
  draw (0,-k) node [right] {New score $score$};
}
end{tikzpicture}
 
end{frame}
  
end{document}