I don't normally post but I also have crippling health anxiety about my babies and my heart absolutely bleeds for you; I know exactly how you feel and that nothing short of definitive medical tests will make you feel better.
I've done some googling so you don't have to and, with my rational head on, it honestly sounds much more likely to be harmless than you're letting yourself believe. In no particular order, here are some reassuring things I read, along with some personal experience.
Firstly it's moveable, and moveable lumps are always a better sign than a fixed one. Usually a moveable lump under the skin is a cyst.
It's also smaller than 1cm. My son had an enlarged node on his neck around that age and the doctor said they're really not concerning unless they're bigger than that.
Supraclavicular lymph nodes are not normally visible except in young children, where they may be. So you may be able to see it because of very minimal swelling because it's just more visible as he's so young.
Lumps in the neck are usually benign. Lumps in young children are much more likely to be congenital than malignant.
Incidentally my mother had an enormous supraclavicular swelling 18 years ago and it was just a fluid filled mass or something. We thought for sure it would be cancerous but it wasn't. Her whole neck was huge.
It's really common for young kids to have coughs that last for months. That in itself isn't even a worry. Same with itchiness, it's winter so skin is drier and there are all sorts of things that could cause itching.
Lymph nodes get bigger and smaller when you have infections or inflammation. Given your boy has symptoms of infection and inflammation, and given that most neck lumps are benign, it really seems that the most likely explanation is that his lymph nodes are up because of a lingering infection.
None of the pictures of supraclavicular lymph nodes I've seen have the lump where you've described it.
The main concern for enlarged supraclavicular lymph nodes seems to be that they are suggestive of being metastases of cancer masses elsewhere in the body, like the abdomen. Obviously no one can say for sure but it seems really unlikely that your baby could have developed a primary cancer elsewhere and progressed to metastatic disease in the very short time he's been alive and you not have noticed any other symptoms before now. By a huuuuge wide margin, the far more likely explanation is that he's been battling seasonal germs and his little lymph nodes are a bit enlarged as his immature immune system fights them off, and you can see and feel them more because he's so tiny and has such delicate skin.
I know you'll be absolutely beside yourself until you get real answers but even to someone who is severely unwell with OCD and health anxiety like me, it really does seem like your baby boy will be just fine.