How can I calculate the potential energy stored by a playing swing?

You can model the equation for calculating the potential energy of the swing, which can be considered a pendulum, as to

$$h=Lcdot(1-cos(theta))$$

Try this link: https://blogs.bu.edu/ggarber/interlace/pendulum/energy-in-a-pendulum/ for further reference.

Work done of a conservative vector field (such as gravity) force, depends only on body path endpoints, it can be stated mathematically as : $$ oint _{C}mathbf {F} cdot d{mathbf {r} }=0. $$ I.e. path integral over closed loop is zero, so total work done is zero. Thus only swing heights matters, it has nothing to do with angles or exact swing path how it travels over time. At each point swing has potential energy : $$ U = mg(h-h_0) $$ where $h_0$ is swing height in it's lowest position relative to Earth surface.