Controlling distance from the arrow with auto=left

I believe you are referring to the labels above the edges/arrows

Each label has been defined as a node -- a node always has some reserved space around it defined by the key inner sep

The key can be set globally at the beginning including the auto key so as to move the label and all nodes close to the edges/ or further

begin{tikzpicture}[inner sep=1pt,auto=left, node distance=2cm,>=latex']
....
end{tikzpicture}

documentclass{article}
usepackage{tikz}
usetikzlibrary{arrows,positioning}

begin{document}

begin{tikzpicture}[inner sep=1pt,auto=left, node distance=2cm,>=latex']
    node at (0,0) (A) {A}; 
    node at (2,1) (B) {B}; 
    node at (3,0) (C) {C}; 
    
    path[->] (A) edge node {x} (B);
    path[->] (B) edge node{y} (C);
    path[->] (C) edge node {z} (A);

end{tikzpicture}

end{document}

An even better option is to define labels as labels rather than nodes -- please see my answer here --

https://tex.stackexchange.com/a/558016/197451

The same code modified (without inner sep invocation)

begin{tikzpicture}[]
    node at (0,0) (A) {A}; 
    node at (2,1) (B) {B}; 
    node at (3,0) (C) {C}; 
    
    path[draw] (A) edge[->] node [pos=0.5, above,  sloped,](){x} (B);
    path[draw] (B) edge[->] node [midway, label={[label distance=-13pt, 
    rotate=45]135:$y$}](){}(C);
    path[draw] (C) edge[->] node [midway, label={[label distance=-6pt, 
    rotate=0]-90:z}](){} (A);

end{tikzpicture}

With TikZ, a node is a box that contains text. This default box is rectangular. If it perfectly frames the text, the edges of the box touch the text, which is unattractive.

By default TikZ leaves a space between the text and the edges of the box which is defined by the length ìnner sep (initially is 0.3333em).

documentclass{article}
usepackage{tikz}
usepackage{amsmath}
%tikzset{every node/.style={draw}}
begin{document}

begin{tikzpicture}
begin{scope}
    node at (0,0) (A) {A}; 
    node at (2,1) (B) {B}; 
    node at (3,0) (C) {C}; 
    
    path[->] (A) edge node[auto=left,draw,inner sep=3em] {x} (B);
    path[->] (B) edge node[auto=left,draw] {y} (C);
    path[->] (C) edge node[auto=left,draw,inner sep=0em] {z} (A);
end{scope}


begin{scope}[xshift=6cm]
    node at (0,0) (A) {A}; 
    node at (2,1) (B) {B}; 
    node at (3,0) (C) {C}; 
    
    path[->] (A) edge node[blue,auto=left,draw,inner sep=3em,label={[blue,above]:$text{inner sep}=3em$}](bigx) {x} (B);
    node[inner sep=0pt](smallx) at (bigx){x};
    draw[|<->,red] (smallx.west)--(bigx.west)node[midway,below,font=footnotesize]{$3em$};
    path[->] (B) edge node[red,auto=left,draw,label={[red,above right,align=center]:initially$text{inner sep}=0.3333em$}] {y} (C);
    path[->] (C) edge node[violet,auto=left,draw,inner sep=0em,label={[violet,below,yshift=-5pt]:$text{inner sep}=0em$}] {z} (A);
end{scope}


end{tikzpicture}

end{document}

When the node is a circle, this circle is circumscribed to the rectangular node which is drawn by default.

documentclass[border=5mm,tikz]{standalone}

begin{document}
begin{tikzpicture}[]
node[draw=red] (a) at (0,0){A};
node[draw=blue,circle] (a) at (0,0){A};
node[align=center,font=tiny] at (0,-.8) {same node with textcolor{blue}{circle} and textcolor{red}{rectangular} shape};
end{tikzpicture}

end{document}