It is a reference to A Monster Calls, which Jim Kay also worked on and illustrated. It is apparently "a sort of Green Man".
In short it is a Green Man which was a reference to Jim Kay's work on A Monster Calls. See the bottom half of my answer for an email I received from him that this is the case.
The creature on the front cover of A Monster Calls does also look a lot similar to the one on the dust cover.
I haven’t actually read or seen this either but Jim Kay also worked on the film and again the creature in the film is very similar.
I emailed Jim Kay from his contact form on Creepy Scrawlers Ltd and this is the response I got.
I've just noticed this creature on the Dust Cover of Harry Potter and the Chamber of Secrets (Illustrated Edition).
I've also seen that apparently it was revealed at some point what it was (potentially in The Prisoner of Azkaban illustrated edition but I can't see if it is).
Have you revealed what it is and if so what is it?
It's a sort of Green Man, I illustrated A Monster Calls, which kind of changed my life really. Wouldn't be illustrating today if it wasn't for that book by Patrick Ness. I worked on the film too. It was just a little nod to that really.
There are other things people haven't noticed yet. On the Chocolate bar wrapper in the chapter opener inside Prisoner of Azkaban, there is a black dog hidden among the hares. Best Wishes