Welcome to the Bunnings Workshop community @DinahW. It's great to have you join us and many thanks for your question.
As someone who is a long term renter myself, the advice I would give to an owner vs a renter would be different. As a renter, your first priority is to report the wear/damage to your property manager. If this is wear and tear then the owner would appreciate knowing about it so they can fix the issue before it causes more damage to their property.
If you have inadvertently damaged the top causing this issue, then there are limited and really only band-aid solutions to this problem. You could fill in the timber area with CustomPine 12.5ml White Melamine Mend Paint which will colour the area white. Repainting the entire top with Dulux Gloss Renovation Range Tiles And Benchtop could also be an option. Please be aware neither of these products will fix an underlying water egress issue. If water is entering the benchtop elsewhere causing these holes then that issue needs to be solved.
As I mentioned previously the first place to start is with the property manager. By repairing or concealing this damage now, you could cause further more extensive repairs to be needed in the future which will then be your responsibility.
Please let me know if you need further assistance or have any questions.