Simple way is with two-part epoxy wood filler. Helps to have a bit of wire frame in there as cork is a bit tricky to get strong adhesion. Here's the idea, take any light wire (needle, guitar string, something stiff) and insert as far into the remaining cork as you can get (probably 1/2 inch). The yellow lines sticking out are your "frame" and will help immensely with holding the filler in place over time.
You
might just press the putty in there and it may hold very well, but the "anchor wire" is the best insurance you'll get against peel out/fall out against bare metal and cork. Secure the patch with some skeleton underneath, it'll hold far better. Small area to work with, but no more daunting than small flytying or hook removal moves in a fish. If you can, when cleaning up the loose edges, undercut the edges a bit like an eroded creek bank and you'll maximize the result. Think pointy exacto knife and some patience.
Removing the entire ring is another option, it would require more material to patch or replace.
If you are asking this question, I'm not going to go into replacing the cork ring or using a decorative string arbor because I don't think you have the supplies. There's other ways of doing this, but it gets more complicated and you state it's an "inexpensive pond rod" so I assume simplicity and functionality is the path here. Fill and sand to shape, quick and easy.
I've used this under skeleton (with deck screws, nails, etc) technique on exterior doors and believe me, it holds the modern epoxy putties in place very well through a lot of stress. Don't just rely on surface contact/adhesion, it'll fail much sooner with patches like that.
