I only have limited knowledge of this and will probably make a false assumption somewhere in this question. Hopefully what I'm trying to ask is clear and someone can offer corrections.

I understood from watching the movie that Captain Marvel gets her super-powers from the Tesseract. Now that Thanos has broken the Tesseract to get the Infinity Stone that was inside, and claimed that stone in his gauntlet, might Captain Marvel's powers be affected? Why or why not? If not, why are her powers no longer tied to the tesseract?

No. The main part that I think you are misunderstanding is that her powers were derived from a power source (the warp drive) made by the Tesseract, but not the Tesseract itself or even the Space Stone, just a source of power made by the Space Stone.

This is kinda similar to how the Maximoff twins gain their powers from Loki's scepter but still retain them perfectly fine even after Ultron breaks it open to acquire the Mind Stone from inside. In other words, the origin of a person's superpowers doesn't have to keep existing to be the source of the powers. Just like the radioactive spider doesn't have to stay alive forever for Spider-Man to keep swinging around.

No

In Avengers: Endgame her powers seem as they were in Captain Marvel and after a few more years honing her skills they seem stronger even.

Almost certainly not

Her powers come from the Space Stone inside of the Tesseract not the Tesseract itself so breaking the Tesseract shouldn’t really have any affect on her powers. She also absorbed the powers of the Infinity Stone and doesn’t keep drawing power from there per my understanding so again any further affects on the Stone shouldn’t affect her.

Lastly, she somehow manages to get to Earth in the Avengers: Endgame trailer and seeing as her powers involved flight through space she almost certainly gets there using her powers. This is how she leaves Earth at the end of Captain Marvel so it is pretty plausible. After having seen the film this is indeed how she gets to Earth.

From what we know of the Infinity Stones, both from the comic and cinematic universes, they’re kind of intrinsic features of reality. The Space Stone took the form of the Tesseract at one time, but anything the Tesseract can do (like providing Captain Marvel’s power) is due to the Space Stone, which still exists as part of the Infinity Gauntlet.

Presumably Thanos, wielding the Gauntlet, could take away Captain Marvel’s powers (or indeed do pretty much anything else), but he’d have to consciously choose to do so. Clearly, since the six Infinity Stones collectively control the whole universe, they don’t stop everything they were doing when they come under Thanos’s control, as that would imply the universe would end the moment he put on the Gauntlet.

(In fact, without giving away potential movie spoilers, this is key to the plot of the Infinity War comics – things are going on “inside” the Stones, even when Thanos has the Infinity Gauntlet)

I think the implication from the movies so far is that the Stones themselves are the sources of power. The Tesseract — like the Orb which contained the Power Stone in Guardians of the Galaxy, Doctor Strange's amulet with the Time Stone in it, or Loki's staff and Vision holding the Mind Stone — was apparently just a container for the Stone.

As such, Carol's powers presumably had nothing to do with the Tesseract itself, and rather came from the presumably-Space-Stone-y energy released when the light speed drive exploded.

It's unclear whether they're dependent on the continued existence of the Space Stone. I don't think there's any particular suggestion of it, but it's possible. As alluded to in Virusbomb's answer, Wanda Maximoff is (or was) our other living example of someone who got powers from an Infinity Stone. Again, there's no indication that she lost her powers after she (temporarily) destroyed the Mind Stone, but it's possible.

No. They don't.

The tesseract was a sort of ... interface, that was built to be able to use the Space stone inside. Much like how the scepter was an interface to the mind stone.

Her energy originates from that stone. Not the casing around it. Thanos didn't break the space stone itself. He just broke its casing open.

Similarly, Wanda and Pietro didn't lose their powers when the Scepter was cracked open to get the mind stone within.