Your textured ceiling complicates the repair. It looks like the drywall has separated from the framing. You need to push it up tight to the framing and add some drywall screws to hold it in place. Then the joint needs to be retaped and mudded. Pull the old tape off to the point where it becomes solidly connected to the drywall. Scrape the popcorn off around the joint and apply tape and drywall mud. Here is a video on the subject:
They make a spray on texture in a aerosol can that can help match the ceiling. Matching textures can be really tough, so it may always look like a patch, but it will look better than what you have today. Alternatively, you could scrape the entire ceiling in this room and apply another coat of drywall mud to have a smooth ceiling. This is a fair amount of work and difficult for a novice to get a smooth ceiling.
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