How to draw a cycle graph with the help of Tikzit?

I wanted to try using "Tikzit" to create a cycle graph. But I have a little trouble. Software download link

It may be easy to use the existing tikz code.https://texample.net/tikz/examples/cycle/

% A simple cycle
% Author : Jerome Tremblay
documentclass{article}
usepackage{tikz}
begin{document}
begin{tikzpicture}

def n {5}
def radius {3cm}
def margin {6} % margin in angles, depends on the radius

foreach s in {1,...,n}
{
  node[draw, circle] at ({360/n * (s - 1)}:radius) {$s$};
  draw[-, >=latex] ({360/n * (s -1)+margin}:radius) 
    arc ({360/n * (s - 1)+margin}:{360/n * (s)-margin}:radius);
}
end{tikzpicture}
end{document}

When I want to use Tikzit software, it seems difficult to draw above graph.

The key lies in the location of the vertex and the setting of the curvature of the curve, which is very difficult for me. Don’t know how to adjust them.

The following codes come from this software.

documentclass{article}
usepackage{tikz}
usetikzlibrary{backgrounds}

usetikzlibrary{arrows}
usetikzlibrary{shapes,shapes.geometric,shapes.misc}

% these are dummy properties used by TikZiT, but ignored by LaTex
pgfkeys{/tikz/tikzit fill/.initial=0}
pgfkeys{/tikz/tikzit draw/.initial=0}
pgfkeys{/tikz/tikzit shape/.initial=0}
pgfkeys{/tikz/tikzit category/.initial=0}

% standard layers used in .tikz files
pgfdeclarelayer{edgelayer}
pgfdeclarelayer{nodelayer}
pgfsetlayers{background,edgelayer,nodelayer,main}

% style for blank nodes
tikzstyle{none}=[inner sep=0mm]

% Node styles
tikzstyle{white_node}=[fill=white, draw=black, shape=circle]
tikzstyle{blackedge}=[-, draw=black]

begin{document}
    begin{tikzpicture}
        begin{pgfonlayer}{nodelayer}
            node [style={white_node}] (0) at (5, 6) {};
            node [style={white_node}] (1) at (7, 4) {};
            node [style={white_node}] (2) at (5, 2) {};
            node [style={white_node}] (3) at (3, 3) {};
            node [style={white_node}] (4) at (3, 5) {};
        end{pgfonlayer}
        begin{pgfonlayer}{edgelayer}
            draw [style=blackedge, bend left] (0) to (1);
            draw [style=blackedge, bend left] (1) to (2);
            draw [style=blackedge, bend right=330] (2) to (3);
            draw [style=blackedge, bend left=45, looseness=0.75] (3) to (4);
            draw [style=blackedge, bend left] (4) to (0);
        end{pgfonlayer}
    end{tikzpicture}
end{document}

Edit: After a reminder from SebGlav , my question can actually be equivalent to this:

How to use TikZ code to do that circle graph with proper edges?

The circle is only part of my drawing, in fact I want to draw the following figure.

documentclass[tikz,border=5mm]{standalone}
begin{document}
begin{tikzpicture}[thick]
defr{2} 
draw (0,0) circle(r)
(0,0)--(90:r)
(0,0)--(150:r)
(0,0)--(-30:r);

foreach point in {(0,0),(90:r),(150:r),(210:r),(30:r),(-30:r),(-90:r),(150:r/3),(150:2*r/3),(-30:r/2),(90:r/2)}
draw[teal,fill=white] point circle(.1);

path
(0,0) node[below=1mm]{$omega$}
(-30:r) node[below right]{$u$}
(-30:r/2) node[above=1mm]{$P'$}
(90:r) node[above=1mm]{$z$}
(90:r/2) node[left=1mm]{$P$}
(150:r) node[left=1mm]{$y$}
(150:2*r/3) node[below=1mm]{$x$}
(-90:r) node[below=1mm]{$C$}
;
end{tikzpicture}
end{document}