Minimum changes
First of all you should add a \clearpage before \end{appendices}, otherwise changes of appendices will end before the last page of the appendix finishes. After this you could additionally use \@chapapp instead of \chaptername. \@chapapp is \chaptername in the main part and \appendixname in the appendix. So
\documentclass[a4paper,10pt]{report}
\usepackage[utf8]{inputenc}% should not been needed with LaTeX >= 2018/04/01
\usepackage{fancyhdr}%Add chapter name at the beginning of each page
%\pagestyle{fancy}
\setlength{\headheight}{15pt}% ...at least 51.60004pt
% Title Page
\usepackage[titletoc,title]{appendix}
\title{}
\author{}
\begin{document}
\maketitle
\begin{abstract}
\end{abstract}
\pagestyle{fancy}
\renewcommand{\chaptermark}[1]{\markboth{#1}{#1}}
\fancyhead[R]{}
\makeatletter% Needed because of @ in macro name → https://tex.stackexchange.com/q/8351/277964
\fancyhead[L]{@chapapp\ \thechapter\ --\ \leftmark}
\makeatother% because of \makeatletter above
\chapter{First chapter about happiness}
First page about the argument
\newpage
Second page about the argument. I want that this page has the chapter name at the top of the page.
\chapter{Hello}
\begin{appendices}
\chapter{Hi}
Appendix about something
\newpage
I do not want that this page has the chapter name at the top of the page.
\clearpage% Dont forget this!
\end{appendices}
\end{document}
results in:

which is consistent with the headings in the main part. To additionally omit the Appendix in the head only, you can use:
\documentclass[a4paper,10pt]{report}
\usepackage[utf8]{inputenc}% should not been needed with LaTeX >= 2018/04/01
\usepackage{fancyhdr}%Add chapter name at the beginning of each page
%\pagestyle{fancy}
\setlength{\headheight}{15pt}% ...at least 51.60004pt
% Title Page
\usepackage[titletoc,title]{appendix}
\title{}
\author{}
\begin{document}
\maketitle
\begin{abstract}
\end{abstract}
\pagestyle{fancy}
\renewcommand{\chaptermark}[1]{\markboth{#1}{#1}}
\fancyhead[R]{}
\fancyhead[L]{\chaptername \thechapter\ --\ \leftmark}
\chapter{First chapter about happiness}
First page about the argument
\newpage
Second page about the argument. I want that this page has the chapter name at the top of the page.
\chapter{Hello}
\begin{appendices}
\fancyhead[L]{\thechapter\ --\ \leftmark}% Change the header for this appendix
\chapter{Hi}
Appendix about something
\newpage
I do not want that this page has the chapter name at the top of the page.
\clearpage% Don't forget this, or the \fancyhead inside appendices will not change the last page of the appendix
\end{appendices}
\end{document}

But I won't do this, because it is inconsistent using Chapter in the main part but not Appendix in the appendix.
Recommended changes
Note, using \the… number in the header definition instead of adding it using \markboth and \markright is always dangerous. It can become valid to early and invalid to late and therefore result in wrong running heads. So I would recommend, to move the counter to the definition of \chaptermark:
\documentclass[a4paper,10pt]{report}
\usepackage[utf8]{inputenc}% should not been needed with LaTeX >= 2018/04/01
\usepackage{fancyhdr}%Add chapter name at the beginning of each page
%\pagestyle{fancy}
\setlength{\headheight}{15pt}% ...at least 51.60004pt
% Title Page
\usepackage[titletoc,title]{appendix}
\title{}
\author{}
\begin{document}
\maketitle
\begin{abstract}
\end{abstract}
\pagestyle{fancy}
\makeatletter% Needed because of @ in macro name → https://tex.stackexchange.com/q/8351/277964
\renewcommand{\chaptermark}[1]{%
\markboth{@chapapp \thechapter\ --\ #1}{@chapapp \thechapter\ -- \ #1}% chapter/appendix name and number moved from \fancyhead[L]{…}
}
\makeatother% because of \makeatletter before
\fancyhead[R]{}
\fancyhead[L]{\leftmark}% chapter/appendix name and number moved to \chaptermark
\chapter{First chapter about happiness}
First page about the argument
\newpage
Second page about the argument. I want that this page has the chapter name at the top of the page.
\chapter{Hello}
\begin{appendices}
\chapter{Hi}
Appendix about something
\newpage
I do not want that this page has the chapter name at the top of the page.
\end{appendices}
\end{document}
As you can see, here you do not needed the \clearpage any longer. Moreover, usually not numbered headings of the table of contents, the bibliography etc. will automatically also not show the prefix and the number. However, if you do not want the prefix (but only the A) in the appendix, you would still need a change, after \begin{appendices}:
\documentclass[a4paper,10pt]{report}
\usepackage[utf8]{inputenc}% should not been needed with LaTeX >= 2018/04/01
\usepackage{fancyhdr}%Add chapter name at the beginning of each page
%\pagestyle{fancy}
\setlength{\headheight}{15pt}% ...at least 51.60004pt
% Title Page
\usepackage[titletoc,title]{appendix}
\title{}
\author{}
\begin{document}
\maketitle
\begin{abstract}
\end{abstract}
\pagestyle{fancy}
\makeatletter
\renewcommand{\chaptermark}[1]{%
\markboth{@chapapp \thechapter\ --\ #1}{@chapapp \thechapter\ -- \ #1}% chapter/appendix name and number moved
}
\makeatother
\fancyhead[R]{}
\fancyhead[L]{\leftmark}% chapter/appendix name and number moved
\chapter{First chapter about happiness}
First page about the argument
\newpage
Second page about the argument. I want that this page has the chapter name at the top of the page.
\chapter{Hello}
\begin{appendices}
\renewcommand{\chaptermark}[1]{% locally redefining the mark to remove the chapter/appendix name (instead of locally changing \fancyhead[L}{…})
\markboth{\thechapter\ --\ #1}{\thechapter\ -- \ #1}%
}
\chapter{Hi}
Appendix about something
\newpage
I do not want that this page has the chapter name at the top of the page.
\end{appendices}
\end{document}
chapappin the heading of your last example is, because you've removed\makeatletterand\makeatotherfrom my examples. – cabohah Feb 15 '23 at 17:54\clearpagebefore\end{appendices}as I stated as minimum requirement, nor moved\@chapapp\ \thechapterfrom\fancyheadto\chaptermarkas I've recommended in my answer. – cabohah Feb 15 '23 at 17:56