This looks like an infection has developed in the gums in that region. Please provide any medical history (illnesses) or allergies that your husband might have so medications can be prescribed accordingly. You will need to get an X-ray if the jaws.
Next Steps
Visit an Oral and Maxillofacial surgeon at the earliest for removal of remaining pieces of tooth and cleaning out of the infection
Get an X-ray.
For right now-
Betadine mouth rinse for gargles 2-3 times a day
Tablet Dolo 650 for pain (since I do not know about your husband’s overall health)
Health Tips
Do not try to press it. Eat food from the other side of the mouth