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The best move would be to simply make sure the existing vapor barrier product is installed properly (flush to ceiling below) and add blown insulation over top to to get the best R value and at the least expense.
Depending on what year your house was built, you might have asbestos in that popcorn ceiling. Typically, anything older than 1979 is a good chance that it could have asbestos; our company regularly finds that with popcorn ceilings. We always send a environmental hygenist out to our jobs to confirm whether or not a house is "hot" for asbestos. It's preventative protection for our workers.
Removing popcorn ceilings requires a lot of preparation to prevent contamination of the other areas of your home. You will need to cordon off the area (room) with some kind of plastic covering the walls, floor, and entry area - this is for your own protection. Also, you will need to wear personal protective equipment including a tyvex type of suit that covers you from head to toe, gloves, safety glasses, and most importantly - a respirator. These things are necessary to keep you safe from breathing in the microscopic fibers of asbestos and preventing you from getting asbestosis or mesothelioma (a tumor of the mesothelium, often malignant and thought to be caused most commonly by the inhalation of asbestos particles.)
If it is a small repair, you may want to test out a product at Lowes. Homax ceiling texture. I'd suggest trying it on a scrap piece of drywall (make sure it's at ceiling height for testing). Jayme and Phillip both had great ideas. Perhaps yet another would be to just remove the popcorn all together.
Typicaly most repairs are noticable and replacement it best. If you simply need to cove the hole and this is a typical way you keep puppy indoors, you may consider purchasing a piece of metal to cover the hole and prevent more damage. Home Depot #800537 is a quick idea.
The caulking mentioned is fine to seals the gape but you may need somethingto back it other thatn backer rod. in that case you may need a small amount of Rockwool insulation to pack into the void befor applying the caulk. This is a link to the product https://www.rockwool.com/applications/exterior-walls/firestopping/
If you only need a small amount check with a local insulation company to see if they will sell or even give you the smaill quanity you need.
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