Simple fix suggestions needed for damaged plywood paneling
I don't think you really have plywood there.  It looks like some OSB type mix with with a veneer - these were super popular in the 60s-80s.   With the veneer off I am afraid that painting it or anything else is going to look bad and not last.   Cardboard doesn't hold paint well.

Since it is a laundry I would go with wainscoting.  They make some super durable ones now that are thin and are almost plastic like.   You could glue these on, add a line of trim above them and the room will look better and be more durable.   

I would also just think about drywalling the room too.   If the rest of your house is drywall it might be an expectation if you were ever selling.