How do I get paint to stay on kitchen varnish on cabinet

you need to seal the paint. Use a brush on polyeurothane. Make sure you have plenty of Air flow and don’t breath the fumes

Or you can use minwax Rub on wax to seal them. Kind of like waxing a car.

They sell an undercoat to apply before you paint, its available at home depot ,lowes ,or any paint store.

Before you paint clean your cabinets with TSP. It will remove any oil or dirt and allows the paint to adhere to even varnished surfaces. I've used on bathroom and kitchen cabinets.

Paint with a primer first.

Now that you have this problem and assuming it is just spot...… and don't want to strip everything. Try putting a little paintable chaulk over it and see if paint will adhere. worth a try.

you need to clean the cabinet first, lightly sand, clean off the dust then paint with a primer based paint

At this point you will need to remove as much of the paint as possible. Lightly sand to roughen for paint. Then you need to prime with a stain blocking primer like Kilz. Then you can paint. No matter what I paint I always prime. Primer creates a bond base for paint. It also makes the finish paint more vivid and color true.