Removing extra space with \pmod command

I'd define a variant of \pmod:

\makeatletter
\let\@@pmod\pmod
\DeclareRobustCommand{\pmod}{\@ifstar\@pmods\@@pmod}
\def\@pmods#1{\mkern4mu({\operator@font mod}\mkern 6mu#1)}
\makeatother

You can use \pmod as before in all other situations and write

\[ \sum_{n \equiv 1 \pmod*{k}} ... \]

when you need it as a subscript to a sum.

Another way might be to load amsmath and redefine \pod (\pmod is defined in terms of it):

\usepackage{amsmath}
\makeatletter
\renewcommand{\pod}[1]{\allowbreak\mathchoice
  {\if@display \mkern 18mu\else \mkern 8mu\fi (#1)}
  {\if@display \mkern 18mu\else \mkern 8mu\fi (#1)}
  {\mkern4mu(#1)}
  {\mkern4mu(#1)}
}

This will avoid having to say \pmod* and the result will be the same as shown.