How do you repair a small damaged spot in a vinyl or linoleum floor?
The only repair I am aware of is to cut out the damaged area and replace it with a new piece. Do you have any scraps of the flooring available? Or a hidden place you can cut out a square (such as under the refrigerator or dishwasher)? However, you must be aware that if your floor is completely glued down, you may not be able to get a clean patch up from the floor. You won't know until you try.
If the floor has a pattern, you have to match it, and the scrap should be larger than the total repair area you are going to cut out. If the floor has a busy pattern, you can try to hide the repair by making cuts over the lines in the pattern.
First, line the scrap piece over the damaged part, so that the patterns align, and tape it down with masking tape.
Then, cut through both pieces together, keeping the sharp utility knife blade as vertical as you can. A carpenters square or any metal straightedge will help you to get the cut right. If you have enough scrap to practice your cutting first, even better. By cutting through both pieces at once, you will cut out the damaged piece and make a matching patch at the same time.
After cutting, remove the damaged piece by carefully inserting a putty knife under a corner and pulling it up. If there are any lumps of glue, scrape them off.
Test fit the patch you cut, doing any trimming until it fits neatly in its future home. Then glue the repair in place. Press the repair patch firmly to get even glue distribution. You can use a folded towel for this step.
There are special sealers that are applied to the seams in sheet flooring. It would be advisable to apply a seam sealer around the perimeter of your patch to prevent moisture from getting beneath it during cleaning or in the event of spills.