hello and welcome to HCM,
In severe infections of the tonsils and also infections where you have not aggressively treated the infections, some amount of pus can come out of the tonsil. Like you rightly mentioned it is called an abscess
and in technical terms its called quincy if it is acute.
In any case you need to consult an ENT to get to a proper diagnosis. You may further require a throat swab for culture and sensitivity to get a better and more directed treatment.
If your pain is more you can consider taking injectables for better symptom control in consultation with your ENT
However if the symptoms are recurring and since you are getting recurrent such lesions it is advisable that you consult an ENT who may advice treatment which can be in the lines of medication or even removal of tonsil which can cure the problem
get well soon
take only warm fluids and do salt water gargle
Dr Sriram Nathan