There is not a practical way to fix that and you are likely to make it worse by trying.
If the hole is only in a thin surface film, you might be able to press it flat, but if the film has been distorted (stretched), it probably won't stay flat. You might find that over time, the film may flatten a little on its own. I would not try to soften it with heat because some plastics will shrink and shrivel, making it worse and eliminating any chance for improvement.
If it is a puncture hole that extends into other plastic layers, you will not be able to flatten the raised rim of the hole without damaging a bigger area.
In terms of filling the hole, that is also likely to make it worse and if not, probably won't improve it. It also depends on the purpose of the surface film and the cause of the bright spot. If it is an anti-glare film,the bright spot may be the next layer. Anything that is not anti-glare film will be a similarly non-diffusing surface. If the filler doesn't have the same refractive index as the next layer, it may create cloudiness or distortion that will also be noticeable.
Filling the hole with something like what is used to fill holes in car windshields is likely to make it much worse. That material is similar to superglue. It may etch the surface. It may wick under the film and damage a large area. The fumes may etch the surface over a large area.
Similarly, trying to glue down the surface film may wick under the film and mess up a large area. If you use an adhesive that doesn't wick, you would need a way to compress the film flat until it cures to avoid a permanent bump. Material thick enough not to wick is likely to leave a raised area. The screen sandwich is manufactured using tightly stretched sheets of material that are compressed together and bonded under pressure. It is not a condition you can reproduce to repair a spot.
If you aren't satisfied with your repair result, you won't be able to undo it without making it worse than it was when you started.
There isn't a practical way to actually repair it, but can you make it less noticeable? A number of people responded with ideas based on the principle that a dark spot may be less noticeable than a bright spot. You could potentially make it a little less noticeable, but it's a question of how much improvement can you achieve and at what risk.
No matter what you do, it will still be noticeable. Maybe a darker spot won't bother you as much if you get it right. But there is a good chance of achieving little or no net improvement, and a substantial risk of making the situation worse in a non-fixable way.
Anything hard, like a pencil or pen tip, can nudge more film loose, making the hole bigger. Any liquid can wick under the film, leaving a big stained area. Assume permanent markers that will stick to the film will be permanent, whether or not it turns out to be an improvement.
Don't count on a redo or cleanup, because cleaning fluids, including any dissolved materials, can wick under the film, and rubbing is likely to make the hole bigger. So if you don't get it as good as you're going to get it on the first try, you have a good chance of making the problem permanently worse.
Now that you've been forewarned, if you are determined to try this, here's an approach I would try if it was a last resort (disclaimer, I've never tested it, but it seems like the least risky alternative). Practice the procedure first on something else to get the feel for how things behave and how much working time you have. Work with a magnifier and good light so you can see what you're doing. Do the procedure with the screen lying face up and level.
Use a very viscous sticky material with temporary adhesion, like rubber cement for paper. Use an extremely fine-tipped brush, or be extremely gentle with a toothpick, to apply a thin layer to the hole, being careful not to get it anywhere else (not getting it anywhere else is more important than perfectly filling the hole).
While it is still tacky, dust on some dry powdered graphite using a pinpoint applicator (sold in hardware stores as a lock lubricant). Use a soft brush and blowing, while protecting the hole, to remove any graphite that lands outside the hole (you can clean the rest of the screen as you normally would, just be careful to avoid the hole). If you've made the problem worse, you may be able to carefully peel this filler off when it dries.
If the hole bothers you so much that you are ready to replace the screen or buy a new monitor, you don't have much to lose by attempting these measures (other than possibly not having the monitor as a backup in the latter case). Otherwise, consider whether the risks outweigh the minor potential improvement.
Unfortunately, this is one of those problems where the best solution may be to change how you view the problem. You're aware of the hole, which serves as a constant reminder. Instead of letting the hole bother you, think about how much money you will save by simply living with it. Every time your eye is drawn to it, remind yourself of the savings from not buying a replacement monitor. :-)