How to repair little slits and holes in vinyl flooring?

Here is an article that might be helpful - https://www.familyhandyman.com/floor/vinyl-flooring/repairing-vinyl-flooring/view-all - good luck

If the vinyl is made to look like wood or stone, get a furniture repair kit. Or you need permanent markers in wood tones/stone tones. Some are made specifically for mending scratches. Even though this is not real wood, you can use wood putty to fill in nicks, cuts and holes. First, clean every bit of the vinyl with warm water and just a few drops of Dawn. Use an old toothbrush to clean out the damaged spaces, then work a bit of the putty in each space. Use an old credit card or a little putty smoother, which looks just like a blank credit card, to remove any excess putty. Use your hand to see if the area feels smooth and add or remove putty as you need to. Now, you wait. Let the putty dry and cure, according to the directions. Once dry, use the permanent markers to color and blend the putty so that it seems to be part of the vinyl. (You can use regular permanent markers; it's just hard to find the right colors.) You now need to seal the repaired areas. I'd test in an obscure area, but a dot of clear fingernail polish should do the job. Once the sealer is totally dry, wax or mop on what you have been using for shine, like Clean 'n Shine, or Mop and Glow. Keep everybody off the floor for at least thirty minutes to an hour. You want a really dry sealer to protect from more damage as long as possible. Polish once a week. Bless you 😇

I really have no idea but to say if you have some area's that have split and lifted perhaps carefully stick them down with super glue or a similar product.