Having worked in with and in the flooring industry some of the advise is right on. !/4" "marine grade" ply wood is the besy underlament that you can use as it has no voids like the ply you get at homedepo etc. it is the standard for flooring. There may have been some misunderstanding on what underlayment is,it is used on top of the SUB floor to get a smooth surface and to allow removal if a floor needs replacing. The type of flooring you have is what is called "Rotoroveer" ( don't know the correct spelling.It is a vinyl photograph film that gives the floor its look/pattern and comes in various grades/thicknesses. The other type is an inlay where the pattern goes all the way through. There may be other sub categories but those are the two I was most familiar with.
If the damage was real bad one poster recommended scribing a piece to repair which is the right way to do it. But to go a step further they make a product that would weld the seams together after you made your patch,and this is what you need to permenantly fix your tear. Any flooring store should have a kit that would do this. Congoleum and Armstrong make such a product.
