It looks like what you have there is Ligustrum japonicum, Japanese Ligustrum. There are many varities of Ligustrum, the most common are Japanese and Chinese (which has a very small leaf - usually 1 inch or less). In my experience with them, the scars don't "heal" well. Most likely you'll always have those large cuts, especially if it's grown in a pot. They can be hollowed in time though and can add an interesting look of age.
I'm sure you have different growing conditions than I have, but I've found that if I don't repot ligustrums every 1-2 years, they go into decline. I don't know if it's a fungal problem, restriction of the roots they don't like, or what. But mature and healthy plants will start dropping branches and eventually die. I've learned I can't be lazy in repotting this species here.
By the way, both the Japanese and Chinese Ligustrums are considered invasives here in Central Texas and are frequently "culled" out of our woodland. They're still sold in garden centers though, and I still really like the species. It's kind of like a temperate climate Ficus.
