This may be a cyst or infection of some sort. There is no way to tell you what this might be. I doubt it is contagious but, if it does not look better, you should have it checked by your Vet.
As Margot said, it may be a cyst or abscess. I don't know where you live, or what diseases are present in your area, but if it is a cyst or abscess, then it isn't contagious. But do be careful to wash your hands with hot water and soap after handling it ir bathing the wound, etc, as sometimes infections like this can cause stomach upsets, if the bacteria are transferred to your mouth or food etc.
At one month, he is really young and tough in his little way...yet also very vulnerable. So you want to be sure the infection has cleared properly and as quickly as possible, so it won't spread to his system. If it does, because of his age it could drag him down very quickly and even possibly kill him.
Some abscesses can burst, and then start to heal over again, and we can think they are okay now, and gone. But there are some with what is called a 'sinus' from deeper in the tissues, where there can be a deeper level infection which hasn't cleared.
So I think it is best to get him checked by a vet to be on the safe side.
It might be that he will need an antibiotic to be sure it has cleared up. Ask the vet.