What could a sore bump behind my last molar on the right side of my mouth be?

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The probable diagnosis is pericoronitis. As per your symptoms, the sore bump behind the last molar tooth can be either due to an abscess formation. It can be due to an injury, or it can be due to an erupting wisdom tooth that might be putting pressure over the gums. Another cause can be inflammation of the gum flap in that area caused due to food accumulation between the gum flap and the wisdom tooth. For confirmation, a clinical examination and an X-ray are needed. I suggest antibiotics like Augmentin (Amoxicillin and Potassium Clavulanate) and anti-inflammatory painkillers like Ibuprofen can help. Consult your specialist doctor, discuss with them, and with their consent, take the medicines. Application of numbing gel containing Lignocaine (Lidocaine Topical) over the bump will also help. A warm saline gargle and an antiseptic mouthwash gargle can also be used. You need to avoid hard foods to prevent irritation of the area, keep the site clean, and maintain meticulous oral hygiene. Follow up with the X-ray.