I'm thinking of a planet with one side too hot, and one too cold due to always facing its star. With the planet having a small ring, a "twilight zone" which may be habitable, I was wondering if a moon with some kind of orbit can somehow give this zone a consistent day/night cycle.
The problem with a tidally locked planet is it can't have a big moon orbiting it, and as the previous answers mention it, it would cast a tiny shadow.
But think of this : on your twilight zone, you have 2 sides : one side closer to the sun, when the sun is actually really low on the horizon (always kind-of day), and one side further, with the sun a bit below the horizon (always kind-of night).
Now, if you allow the planet not to be quite totally tidally locked, you can have it slightly oscillating (something like a span of a few degrees). Maybe in a few million years the planet will be effectively tidally-locked, but today is not that day, it's not rotating anymore but still oscillating. This way, the sun-close side of the ring of habitability have a long day / short night cycle, while the sun-away side has a short day / long night cycle.
Of course, it's not really day/night, but more of a dusk/dawn. Think of the course of the sun on the poles, but with day/night cycle of a few hours instead of 6 months.
A moon would provide a very short night.
Note that rings can be arcs not full rings! You'd have to look into that, but just off the top of my head I would suppose that an arc ring would have to be very narrow and thus would ot eclipse the sun anyway. But it’s a type of phenomenon you might not have heard of, and might give you more ideas.
That's a really interesting idea. I'm actually working on a tidally locked planet as well right now, and I had never thought of that. I can't see why it wouldn't be feasible. All I can see is that the moon would have to be really quite large, possibly even the size of the planet itself, for there to be a day/night cycle like we have on Earth. For reference, in the 2006 total solar eclipse, the moon only cast a shadow on a portion of Turkey and Cyprus: https://en.wikipedia.org/wiki/Solar_eclipse#/media/File:Eclipse_fromISS_2006-03-29.jpg
The implications of having a moon that large would most likely drastically change the path of orbit of the planet, the two would orbit around each other as well as their star.