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The other objects themselves move much, much slower than light. So in general, no, we don't see anything weird. There's no way for one object to be in two places over time such that the light from both places reaches us simultaneously, under normal conditions. You'll always see a single image.

There are some exceptions; gravitational lensing can make two images of one object because the light takes two paths, sometimes offset by time. (A supernova we see from one image of a galaxy can appear in another image of it years later.)

Also, the spirals of galaxies that we see nearly edge-on, like Andromeda, are distorted because the light from the far side is delayed by a few hundred thousand years compared to the front. Not exactly smeared, but, that's getting closer to the speeds and scales you'd need to have.

The light would’ve already passed us. We can only see light that hits our telescope every nanosecond and every nanosecond new light from that object reaches us. We can never determine where in the sky that light started to travel. Our snapshot of the universe is exactly that; the light that we see exists in the moment it reaches us locally but the object it originated from is never in the same position as we see it.

For the same reason you don't see such "trails" of the cars on the road next to you. The light coming from the time when object was in different location already reached you in the past.

The "trails" you refer to are essentially what you perceive as "object moving".

Thanks this is the answer I was looking for! The smearing effect is what I was anticipating, I see now why it doesn’t happen for stars and the like.

One interesting example of how the very fast but finite speed of light can change what we see when we look at an object over an extended period of time is the observed light echo in V838 Monocerotis system. In the video (which is composed of individual observations over a time base of 2 or 3 years), it looks like we are watching a shell of debris expand out away from the central pair of objects, but this is actually not the case. Rather, what we are seeing is light traveling outward and illuminating successively further layers of a pre-existing but invisible shell of material

That’s pretty wild. This is the type of stuff I’m talking about. Anyone coming across that without knowing better might assume it’s an active explosion.