Thank you for your query on Healthcare Magic.
The bump at the base of the skull is looking like an inflamed lymph node
as it is tender and movable.
There will be sub occipital lymph nodes at the base of the skull, and whenever there is a scalp infection or some systemic infections like viral and bacterial, they get inflamed and enlarged.
Scalp should be examined to see the presence of any local infections like bacterial or fungal.
or pustule lesions, folliculitis
of scalp will lead to enlargement of lymph nodes in their draining area.
I advise you to consult your Pediatrician
for the examination and for treatment with topical or oral antibiotics
Hope I have answered your query.