Cfrgtkky

--- EDIT 2 ---

I've created the following header using an image editor to help you guys understand what I'm trying to do.

--- End of EDIT 2 ---

--- EDIT 1 ---

This is closer to what I have in my header. Only instead of actually writing "Chapter 1" and "Subsection", I'm using rightmark and leftmark, but the result is basically the same. And, of course, I'm using a different image.
The problem is with the line crossing the image. Or maybe with the length of the header line.

--- End of EDIT 1 ---

Ok, I think I've searched for this everywhere and I can't come up with an answer. Does anybody know how to put a space between the image and the header line, so that the line doesn't cross the image?
This is what I have so far.

documentclass[12pt,twoside]{report}

usepackage{graphicx}

usepackage{fancyhdr}

usepackage{blindtext}

pagestyle{fancy}

% ---------------------------- Header ---------------------------- %

fancyhead{} 

fancyhead[RE,RO]{nouppercase{Subsection 1.1}}

fancyhead[CO,CE]{MakeUppercase{Chapter 1}}

fancyhead[L]{raisebox{-.6height}[0pt][0pt]{includegraphics[width=1cm]{example-image-a}}}

% ---------------------------- Footer ---------------------------- %

fancyfoot{}

fancyfoot[RE,RO]{thepage} 

renewcommand{headrulewidth}{0.4pt}

renewcommand{footrulewidth}{0.4pt}

begin{document}

blindtext

end{document}

A simple solution redefines headrule to fine-tune its placement:

documentclass[12pt,twoside]{report}

usepackage{graphicx}

usepackage{fancyhdr}

usepackage{blindtext}

pagestyle{fancy}

% ---------------------------- Header ---------------------------- %

fancyhead{}

fancyhead[RE,RO]{nouppercase{Subsection 1.1}}

fancyhead[CO,CE]{MakeUppercase{Chapter 1}}

fancyhead[L]{raisebox{-.7height}[0pt][0pt]{includegraphics[width=1cm]{example-image-a}}}

% ---------------------------- Footer ---------------------------- %

fancyfoot{}

fancyfoot[RE,RO]{thepage}

renewcommand{headrulewidth}{0.4pt}

renewcommand{footrulewidth}{0.4pt}

renewcommand{headrule}{hspace*{1.2cm}rule[2ex]{dimexprtextwidth-1.2cm}{0.4pt}}



begin{document}



blindtext



end{document} 

Adding a vspace after your includegraphics should do the trick. No razebox needed:

documentclass[12pt,twoside]{report}

usepackage{graphicx}

usepackage{fancyhdr}

usepackage{blindtext}

pagestyle{fancy}

% ---------------------------- Header ---------------------------- %

fancyhead{} % blank fancyhead command which clears all the header fields.%

% --------- The lines below are commented because I'm using in the original file, but I think that there is no need for it in here. ---------

% fancyhead[RE,RO]{nouppercase{rightmark}}

% renewcommand{chaptermark}[1]{%

% markboth{#1}{}}

%    fancyhead[CO,CE]{MakeUppercase{leftmark}}

fancyhead[L]{includegraphics[width=1cm]{example-image-a}vspace{3pt}}

% ---------------------------- Footer ---------------------------- %

fancyfoot{} %Clears the footer fields using a blank fancyfoot command.%

fancyfoot[RE,RO]{thepage} % makes the page number appear on the right of the footer for an even page and the right for an odd. The thepage command returns the page number of the page it's used on.

renewcommand{headrulewidth}{0.4pt}

renewcommand{footrulewidth}{0.4pt}

begin{document}

blindtext

end{document}