The left hand image above appears to show the surface has been scraped with something, and that the colour seems to be primarily in the existing polish, although the base veneer may also have been dyed. Assuming you are not going to chemically strip the existing polish prior to refinishing then coloured wood scratch repair markers might be able to replace the colour, see here:
https://www.morrells.co.uk/products...repair/mohawk-repair-products/mohawk-markers/ and polish over locally to match.
As to the place where the veneer has been penetrated and broken up, there is also damage to the ground underneath. This will require cutting away the veneer around the damage, filling the hole and then, ideally, cutting a new piece of matching veneer to fit into the hole you created when you cut the damaged veneer away. On the other hand, you might just about get away with cutting the damaged veneer away minimally, filling the hole(s,) and doing a bit of grain matching with marker pens or fine brushes and acrylic paint, then, as above, polish over locally to match.
There's no guarantee of perfection in undertaking such repairs, but hopefully you can improve the situation.
Finally, if you're not going to strip the existing finish off, and your plan is to apply varnish over the top of that finish, plus repairs, it's possible the varnish you have will not be compatible, and therefore may not adhere properly. If that's your plan, you ought to test the compatibility of the varnish with the existing finish at some inconspicuous place, e.g., a small patch on the less seen underside or somewhere similar. Slainte.