How do you create a keyhole contour not centered at the origin?

Question

I've been trying to create a keyhole contour where the keyhole is a tiny circle in say, the first quadrant, and the main circle is still centered at the origin. I'm specifically not trying to create a keyhole contour where the keyhole is at the origin, and can't seem to find how to do that. I've been playing around with it for a while, but can't make anything work. Does anybody have any tips?

Thanks!

Edit: Here's what I'm trying to create:

Answer 1

Like this:

In image is considered $sin{alpha} ~ \alpha$ (at small \alpha) and ratio between angles of hole in locus is about 8:

\documentclass[tikz,
               border=3mm,
               ]{standalone}
\usetikzlibrary{arrows.meta, decorations.markings}

\begin{document}
\begin{tikzpicture}[>=Straight Barb,
decoration={markings,% switch on markings
            mark=between positions 0.2 and 0.8 step 0.25 with {\arrow[thick]{>}},
            mark=at position 0.86 with {\arrow[thick]{>}},
            mark=at position 0.98 with {\arrow[thick]{>}}
        }]
% curve
\draw[fill=gray!10, postaction={decorate}] 
            (47.5:2) arc (47.5:360:2   ) arc (0:42.5:2) 
       -- + (225:0.5) arc (25  :0  :0.25) arc(360:65:0.25) -- cycle;
% singularity
\fill[red]  (45:1.25) circle (1pt) node[inner sep=2pt,font=\footnotesize,right] {$a$};
% radius 
\draw[thick, ->] (0,0) -- node[sloped,above] {$r$} (330:2);
% coordinate axes
\draw[->] (-3,0) -- (3,0) node[right] {$\Re$};
\draw[->] (0,-3) -- (0,3) node[above] {$\Im$};
\end{tikzpicture}
\end{document}