Adding text between Bibliography header and the first bibitem

The natbib package has a \bibpreamble hook. You can populate the text with \renewcommand{\bibpreamble}{<text>}.

\documentclass{amsbook}
\usepackage[numbers]{natbib}
\renewcommand{\bibfont}{\small}

\begin{document}

\renewcommand{\bibpreamble}{First. This and that can be found in~\cite{testa}. 
  The other stuff is particularly well explained in~\cite{testb} 
  and~\cite{testc}.}
\begin{thebibliography}{9}
\bibitem{testa} Test A.
\bibitem{testb} Test B.
\bibitem{testc} Test C.
\end{thebibliography}

\renewcommand{\bibpreamble}{Second. This and that can be found in~\cite{testd}. 
  The other stuff is particularly well explained in~\cite{teste} 
  and~\cite{testf}.}
\begin{thebibliography}{9}
\bibitem{testd} Test A.
\bibitem{teste} Test B.
\bibitem{testf} Test C.
\end{thebibliography}

\renewcommand{\bibpreamble}{Third. This and that can be found in~\cite{testg}. 
  The other stuff is particularly well explained in~\cite{testg} 
  and~\cite{testi}.}
\begin{thebibliography}{9}
\bibitem{testg} Test A.
\bibitem{testh} Test B.
\bibitem{testi} Test C.
\end{thebibliography}
\end{document}

EDIT For the font \renewcommand{\bibfont}{\small} changes the font (size) for the bibliography (but not the preamble).

One possible solution; in the code below I defined a \bibNote command that allows the placement of text after the heading and before the first bibitem; the font for this note can be changed by redefining the \Notefont command (default value \normalsize). A redefinition of \bibname allows to change the heading.

\documentclass{amsbook}

\renewcommand\bibname{References and further reading}

\makeatletter
\newcommand\Notefont{\normalsize}
\long\def\bibNote#1{\gdef\@bibNote{\item[]{\Notefont#1}}}
\renewenvironment{thebibliography}[1]{%
  \@xp\chapter\@xp*\@xp{\bibname}%
  \normalfont\footnotesize\labelsep .5em\relax
  \renewcommand\theenumiv{\arabic{enumiv}}\let\p@enumiv\@empty
  \list{\@biblabel{\theenumiv}}{\@bibNote\settowidth\labelwidth{\@biblabel{#1}}%
    \leftmargin\labelwidth \advance\leftmargin\labelsep
    \usecounter{enumiv}}%
  \sloppy \clubpenalty\@M \widowpenalty\clubpenalty
  \sfcode`\.=\@m
}{%
  \def\@noitemerr{\@latex@warning{Empty `thebibliography' environment}}%
  \endlist
}
\makeatother

\bibNote{This and that can be found in~\cite{testa}. The other stuff is particularly well explained in~\cite{testb} and~\cite{testc}.}

\begin{document}

\begin{thebibliography}{9}
\bibitem{testa} Test A.
\bibitem{testb} Test B.
\bibitem{testc} Test C.
\end{thebibliography}

\end{document}

If babel is being used, the redefinition of \bibname has to be done using \addto\captions<language>; for example, if the document contains

\usepackage[english]{babel}

one has to say

\addto\captionsenglish{%
  \renewcommand\bibname{References and further reading}}

I think this should work:

\renewcommand\bibname{References\\
\normalsize{} Citation data collected from Scopus 
(Elsevier) Sep. 6 2017.}

An example:

\documentclass\[a4paper,12pt\]{report}

\renewcommand\bibname{List of books\\ \vspace{10mm} \normalsize{} Note: Strindberg has also written other books.}

\bibliographystyle{plain}

\begin{document}

One book written by Strindberg is Hems\"{o}borna \cite{book1}.

\begin{thebibliography}{9}

\bibitem{book1}
    August Strindberg, \textit{Hems\"{o}borna}, 1887.
\end{thebibliography}

\end{document}