If its the style you are looking for, clearing out the punky junk in it and leaving it as-is is certainly one way to go.
If you wanted a different look, then you can also look at fillers. Epoxy, including tinted/colored with dyes and pigments, can certainly work. You would need a pressure pot or something like Liquid Diamonds to make an epoxy fill bubble-free.
Another option is wood filler. I generally use Goodfilla as my go-to, however Milliput is also another option. Once the filler has cured, you can turn and sand it down to the surface of the rest of the bowl. I often go for contrasting colors with something like this...so, say, a darker wood filler, but you can certainly find something that is a closer match.
Finally, you could also try a cutout and replacement with another piece of wood. Maybe drill out the area with a forstner, then turn another piece of wood round to fit the hole. Fit the filler piece in the hole, then turn that down to the thickness of the rest of the bowl wall. Again, I would generally opt for a contrasting wood, but, some people will find a piece of wood of the same type that has a close grain match, and you can sometimes hardly tell there was any issue at all. (In your case here with that figure, that might be tough...maybe a piece of darker wood.)
You could also close-cut out the punky area, to preserve the figure. Then use a bandsaw or scroll saw to cut a piece of wood that is a close fit to that shape. File and sand it down to be a perfect fit for whatever shape hole you end up with, then glue it in place and turn it down.