(Bother, I forgot that hitting enter submitted the comment - ah well, you'll get notified of that one and then hopefully see the rest!)

So, most significantly, by default then if you specify shorten >=5pt, shorten <=5pt on a fragmented path then it applies it to the last component only.

\draw[line width=2mm] (0,0) to[out=0,in=180] (2,0) (2.5,0) to[out=0,in=180] (4,0);
\draw[red,shorten <=5pt, shorten >=5pt] (0,0) to[out=0,in=180] (2,0) (2.5,0) to[out=0,in=180] (4,0);

(Second bother: I can't just paste an image directly into the chat, I have to save it and upload it ... it's too late in the evening for that right now so you can run the above code for yourself and see that the red line is shortened only on the second component.)

So when the fancy dashed style is applied to the whole pparrowyy path after it has been split, then only the second component gets the shortening. This means that the gap is width 1.25pt before the intersection point and 6.25pt after it (1.25pt being half of 2.5pt). So either we specify the full desired width initially or we need to ensure that the shorten keys are applied to all components. One way is to render each component separately, as you did.

It may be (need to read the manual) that the shorten keys are affected by the tips keys which would provide another way to achieve the same end.

\documentclass[tikz,border=3pt]{standalone}
%\url{tex.stackexchange.com/a/720532/86}
\usetikzlibrary{spath3,intersections}

\begin{document}
\begin{tikzpicture}[
    fancy dashed/.style={shorten <=5pt,shorten >=5pt},
  ]
  \coordinate (yy) at (6.8,-0.4);
  \coordinate (ww) at (7.75,0.7625);
  \coordinate (pp) at (0.75,-2.25);
  \coordinate (qq) at (2.5,-2.25);
\coordinate (ctrl) at (9,-1.5);
\foreach \cdt in {yy,ww,pp,qq,ctrl} {
  \fill[cyan!50!black] (\cdt) circle[radius=5pt];
  }
  \path[spath/save=pparrowyy,draw, fancy dashed, line width=2mm, line cap=butt ] (pp) to[out=-30,in=23…

In the above code, then I apply the shortening to the original path (in black), and then also to the green path. As you can see, the green path is not fully centred on the black path. This is why I think it best to apply the original shortening to the original path and then use the correct width when inserting gaps. So to get the best approximation of what you originally had, I would put the shorten keys on the original path, and then use 6.25pt in the gaps.

Sorry, 12.5pt since it gets split in twain.

\documentclass[tikz,border=3pt]{standalone}
%\url{tex.stackexchange.com/a/720532/86}
\usetikzlibrary{spath3,intersections}

\begin{document}
\begin{tikzpicture}[
    fancy dashed/.style={shorten <=5pt,shorten >=5pt},
  ]
  \coordinate (yy) at (6.8,-0.4);
  \coordinate (ww) at (7.75,0.7625);
  \coordinate (pp) at (0.75,-2.25);
  \coordinate (qq) at (2.5,-2.25);
\coordinate (ctrl) at (9,-1.5);
\foreach \cdt in {yy,ww,pp,qq,ctrl} {
  \fill[cyan!50!black] (\cdt) circle[radius=5pt];
  }
  \path[spath/save=pparrowyy,draw, fancy dashed, line width=2mm, line cap=butt ] (pp) to[out=-30,in=23…

In the above, then the yellow path goes exactly along the centre of the black path (whereas the green is slightly off), and it ends exactly on the boundary of the circle at the intersection point.

So, in conclusion, shorten should be applied to the original path only, and avoided when rendering components.

@AndrewStacey Ha! I have to do this for all images at the moment because the quality is useless otherwise. Actually, I can't just save the file, even. I have to convert outside the viewer, which sometimes doesn't even work. But, if it weren't for that, I'd agree with you.