Any connection between a nasal cauterization and then face pain following that?

Possibly: I have seen many patients with chronic pain after ENT procedures. See someone experienced with chronic postsurgical pain if an ENT md can not fix it.

A Burning Backlash: Nasal cauterization traumatizes the mucosal lining of the nasal passage which leads to inflammation. The subsequently inflammation leads to nasal mucosal edema or swelling. This is not only a localized phenomenon. As a result mucosal swelling can extend into the sinus cavities of the face, especially the maxillary sinus (below the eye). Facial pressure and pain therefore is not uncommon.

Time frame: Depends on the time frame. If immediate - yes. If later, maybe not. See oral surgeon or orofacial-TMJ doc for assessment/diagnosis/treatment.

Very common: After nasal cauterization, you may feel itching and pain in your nose and face for for 5 to 10 days. If concerned see your doctor for post-op appointment.