Swollen lymph nodes on one year old

Early December we noticed a small bump on the left side of my sons neck and so I took him to the hospital and they put him on antibiotics then after one week it was still there so I took him back they just sent me home again with more antibiotics to last three weeks it’s now finished and the lump is bigger and now red. I took him to the drs today and he has just sent me a referral to see a specialist as an out patient. The lump doesn’t seem to bother him but it just looks painful and as a mother on a one year old it’s just not a nice feeling to see that on my baby and the hospital just keep kicking us out as he still seems “happy”. Any thoughts on what should happen? It looks like ot need drainage which what they mentioned the first time but they said he’d have to be vomiting diarrhoea fainting for them to do any surgery on him but it obviously needs more then antibiotics.

Day one we noticed took him to the hispital

One month later....