Preserve page numbering in fancyhdr

This can be done with the help of the afterpage package.

Before the page where you want the page number begins, put:

\afterpage{\cfoot{\thepage}}

So, your MWE can be rewritten as:

\documentclass[10pt]{article}
\usepackage{setspace}
\usepackage[margin=1.250in]{geometry}
\singlespacing
\usepackage{fancyhdr}
\usepackage{afterpage}
\fancyhf{}
\pagestyle{fancy}
\rhead{\textcolor{gray}{Mock Draft}}
\renewcommand{\headrulewidth}{0pt}
\usepackage[table]{xcolor}% http://ctan.org/pkg/xcolor

\begin{document}
xxxxx
\afterpage{\cfoot{\thepage}}
\clearpage
xxxx
\clearpage
xxxx
\end{document}

Note that I deleted letter from the options of the article class. The default is already the letter paper and anyway the right option is letterpaper.

Using \fancyhf{} clears the header and footer of anything. However, if you wish to insert the page number, just add \thepage to the appropriate location. If you wish to do this conditionally, that is also possible through something like

\fancyhead[L]{\ifnum\value{page}<2\relax\else\thepage\fi}

which will place \thepage int the [L]eft header if its value is at least 2.