A trial: I assume that the left mark should be empty (or it should use another style at least).
The \chaptermark{}
must be redefined at begin of the \document
body, otherwise the standard definition of \@chapapp.~#1
etc. is used.
Please note that the chapter start page is using another pagestyle, i.e. plain
.
\documentclass[oneside]{book}
\usepackage{fancyhdr}
\usepackage{blindtext}
\makeatletter
\AtBeginDocument{%
\def\chaptermark#1{%
\markboth {\MakeUppercase{%
\ifnum \c@secnumdepth >\m@ne
\if@mainmatter
\bfseries\thechapter~--~ %
\fi
\fi
#1}}{}}%
}
\fancyhead[C]{\nouppercase{\leftmark}}
\fancyhead[R]{}
\fancyhead[L]{}
\makeatother
\pagestyle{fancy}
\begin{document}
\chapter{First one}
\blindtext[5]
\section{A section}
\chapter{Next chapter}
\blindtext[5]
\end{document}

documentclass
what#1
might be, so you're right: we need a MWE\leftmark
should appear in the center of the head?\chaptermark
is used with which argument, thus without MWE it is not clear where (or whether) to add\thechapter{} -
(or to change something else).