The following figure is a modification of an answer to Sketching free-hand, importing into TikZ.
The labels $K$ and $U_y$ in the picture designate respectively: the blue-filled region that has a thick solid boundary; and the upper-right-most region, with a dashed boundary, that is partly orange-filled but overlaps region K. Each of these labels is obscured by portions of the boundaries: label $K$ by the dashed lower part of the boundary of $U_y$; and label $U_y$ by the solid boundary of region K.
Question: How can I break those two boundaries so as to allow the labels to be more readily seen?
More specifically, is there some nice way to avoid lots of trial-and-error in locating suitable points along the boundaries at which to stop and restart the bounding curves?
(Note: The final picture will use shades of gray instead of colors red, blue, and orange, as specified by the now commented-out code block with the 3 \definecolor commands. I'm leaving the colors as shown so as to indicate more clearly which region is which.)
\documentclass[tikz,border=0pt]{standalone}
% Based upon answer by @marmot 2018/06/10
% https://tex.stackexchange.com/questions/435746/sketching-free-hand-importing-into-tikz
\usetikzlibrary{calc,intersections,arrows.meta,backgrounds}
\usepackage{pgfplots}
\usepgfplotslibrary{fillbetween}
%\definecolor{red}{gray}{0.15} % medium gray for region V
%\definecolor{blue}{gray}{0.2} % dark gray for region K
%\definecolor{orange}{gray}{0.6}% light shading for left & right sides
\begin{document}
\begin{tikzpicture}[long dash/.style={dash pattern=on 4pt off 2pt},
dot/.style = {circle, fill, minimum size=#1,
inner sep=0pt, outer sep=0pt},
dot/.default = 3pt, % size of the circle diameter
x=0.44cm,y=0.44cm % scale units for overall correct size
]
% Left-hand regions
\draw[thick,long dash,name path=left,fill=orange!30] plot[smooth cycle] coordinates
{(0.3,-2) (-1,-3) (-8,-1.2) (-8.8,-0.2) (-7,0.6) (-1,-0.6)};
\draw[thick,long dash,name path=left bottom] plot[smooth cycle] coordinates
{(-8,-2.8) (-9,-2.5) (-8.5,-1) (-7,0) (-6,1.7) (-5,1.7) (-4,-0) (-5.5,-2)};
\draw[thick,long dash,name path=left top] plot[smooth cycle] coordinates
{(-7.2,-1) (-7.8,1) (-6.7,2) (-5.5,1) (-5,0) (-5.4,-1) (-6,-1.2)};
\path [%draw,blue,ultra thick,
name path=left arc,
intersection segments={
of=left top and left,
sequence={A1--B1}
}];
\path [%draw,red,ultra thick,
fill=red!30,
name path=left blob,
intersection segments={
of=left bottom and left arc,
sequence={A1--B0}
}];
% Right-hand regions
\path[fill=orange!30] plot[smooth cycle] coordinates % region U_{u}
{(-1.3,2) (-0.7,3) (1,3.7) (5.2,3) (8,1.6) (8.4,1) (8,0.3) (6,0) (4,0) (2,0.3) (0,1)};
\path[fill=blue!30] plot[smooth cycle] coordinates % region K
{(0,-2) (-0.3,-1.5) (-0.2,0) (-0.3,1) (-1,2) (0,2.8) (3,2) (7,1) (7.3,-1)
(6,-2.3) (4,-2.3) (2,-2)};
\draw[thick,long dash,name path=right top] plot[smooth cycle] coordinates % boundary of U_{u}
{(-1.3,2) (-0.7,3) (1,3.7) (5.2,3) (8,1.6) (8.4,1) (8,0.3) (6,0) (4,0) (2,0.3) (0,1)};
\draw[thick,name path=right] plot[smooth cycle] coordinates % boundary of K
{(0,-2) (-0.3,-1.5) (-0.2,0) (-0.3,1) (-1,2) (0,2.8) (3,2) (7,1) (7.3,-1)
(6,-2.3) (4,-2.3) (2,-2)};
% boundary of unnamed region to lower-left of point y:
\draw[thick,long dash,name path=middle] plot[smooth cycle] coordinates
{(0,-3.4) (-1,-2) (-1,-0.5) (-1.5,0.4) (-1,1.6) (0,1.9) (2.1,1) (3,-1) (2.5,-3) (1,-3.7)};
% boundary of unnamed region to lower-rightof point y:
\draw[thick,long dash,name path=right bottom] plot[smooth cycle] coordinates
{(1,-3) (0.6,-2) (1.2,0) (3,0.8) (6,0.8) (8.5,1) (10,0) (9,-3) (7,-3.7) (5,-3.6) (2,-3.6)};
\path[name path=circle] (5.2,1.5) arc(-30:190:4mm);
\path [%draw,red,ultra thick,
name path=aux1,
intersection segments={
of=circle and right,
sequence={B1}
}];
\path [draw,ultra thick,
name path=aux2,
intersection segments={
of=circle and aux1,
sequence={B0}
}];
% Distinguished points & their labels
\node[dot] at (-6.5,-0.425) {};
\node at (-6.95,-0.675) {$x$};
\node[dot] at (3.3,1.5) {};
\node at (2.85,1.25) {$y$};
% Region labels
\node at (-5.625,-0.025) {$V$};
\node at (-2.5,-1.5){$V_y$};
\node[] at (3.7,0){$K$};
\node[] at (4.8,1.6) {$U_y$};
\end{tikzpicture}
\end{document}
dash patternto leave a gap.Kthe label of the region, or the label of the path?\node[] at (3.7,0){$K$};to\node[] at (3.7,0)[fill=blue!30]{$K$};for example. Doing the same forU_yis a little harder.$K$off that solid boundary without ambiguity as to the meaning. However, moving label$U_y$off the dashed boundary will create ambiguity, since it refers to a region larger than the right-hand (now) orange-shaded region and is supposed to indicate that the labeled region includes the part of that orange region that "dips below" the blue-shaded region.