You are likely to be having chronic tonsillitis.
In chronic tonsillitis there is a reservoir of bacteria left behind which is difficult to eradicate by antibiotics.
Usually in chronic tonsillitis the tonsils get enlarged but in a particular case called atrophic tonsillitis, the tonsils may not be enlarged.
The vessels that you have pointed out are not significant, such vessels are present in most persons, and do not signify anything. Dont worry about that.
If the tonsils are causing you chronic problems, then you should consider a tonsillectomy.
It is good that you are scheduled to see a ENT.
Discuss about atrophic tonsillitis and discuss about doing a tonsillectomy. Also discuss about doing a throat swab and culture to find out the bacteria on your tonsils so that you can get a targetted approach.
These two points will help you in getting a permanent cure.
Take care- be safe - note down these points and see your doctor and discuss with him/her - this will enable you to get an informed discussion leading to the best possible outcome.
Hope this shows you the way forward - wish you good health and a quick recovery.
I trust my answer has helped you gain more knowledge and confidence about your problem and in knowing what to discuss with your doctor.
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