Place a given symbol inside \cup like \uplus

We need fully scalable fonts for this, so fix-cm if the document is in the standard Computer Modern font. It is not necessary if other fonts are chosen.

\RequirePackage{fix-cm}
\documentclass{article}
\usepackage{amsmath}

\makeatletter
\DeclareRobustCommand{\vcup}[1]{%
  \mathbin{\vphantom{\cup}\mathpalette\vcup@{#1}}%
}
\newcommand{\vcup@}[2]{%
  \ooalign{%
    $\m@th#1\cup$\cr
    \hidewidth
    \raisebox{%
      \dimexpr\depth+\demote@{#1}\dimexpr1.5pt\relax\relax
    }{\fontsize{\demote@{#1}\dimexpr\f@size pt}{\z@}\selectfont\check@mathfonts
      $\m@th#2$}%
    \hidewidth\cr
  }%
}
\newcommand{\demote@}[1]{%
  \ifx#1\displaystyle 0.7
  \else\ifx#1\textstyle 0.7
  \else\ifx#1\scriptstyle 0.5
  \else 0.4\fi\fi\fi
}
\makeatother

\begin{document}

$A\vcup{k}B^{\vcup{k}}$

$\uplus\scriptstyle\uplus\scriptscriptstyle\uplus$
$\vcup{k}\scriptstyle\vcup{k}\scriptscriptstyle\vcup{k}$
$\vcup{y}\scriptstyle\vcup{y}\scriptscriptstyle\vcup{y}$
$\vcup{g}\scriptstyle\vcup{g}\scriptscriptstyle\vcup{g}$

\end{document}

The \stackinset command from stackengine can do that:

\documentclass[border = 2pt]{standalone}

 \usepackage{stackengine} %
\newcommand\cupk{\ensurestackMath{\stackinset{c}{0ex}{c}{0.2ex}{\scriptscriptstyle k}{\scriptstyle\cup}}}

\begin{document}

$B^{\cupk} $

\end{document}