Assuming that the planet is bigger than the moon, then the planet would have a much larger impact on the moon than the moon would have on the planet.

Look at the moons of Jupiter and Saturn. They are squeezed and compressed by the planet's gravity, which causes volcanism. Io is the extreme example, as it is closest to Jupiter. But Ganymede has a liquid ocean under it's ice largely because the gravity of the nearby planet is squeezing it and the friction is warming up the interior.

Also tidal locking is a thing. That's good for your story. A moon that is tidally locked to it's planet (like ours is) will always face the planet. From our perspective the moon doesn't rotate, it's always showing us the same face. That's why the moon has a dark side. Not because it's literally dark, but because before we invented spacecraft no one had ever seen it, it was unknown.

So your theoretical moon would have the city always facing the planet, to remind the planet's inhabitants of what happened (although it would likely be much to far away to make out the city). It could also be a sulphurous volcanic wasteland, much like the surface of Io due to tidal effects.

As for the moon's effect on the planet, that depends on the size of the moon and how close it is to the planet. Too close and too large and tidal effect will break the moon apart. You seem to have settled on a small, close moon which likely would have very little effect on the larger planet. As an example the 2 small moons of Mars have almost no effect on the planet at all.

Our moon causes ocean tides on Earth. It may also have an effect on plate tectonics, not so much with moving them, but may be at least part of what's keeping them apart and not allowing them to bind together (The jury is still out on that one, the oceans being subducted and releasing water seems to play a much larger role). On the other hand, the Earth caused the moon to become tidally locked and also causes a regular pattern of moonquakes.