Yes, horses can definitely fully recover from abscesses *generally*. They can also recover from puncture wounds to the foot, but I suppose it depends how severe the wound was. Good think puncture wounds are quite uncommon.
Abscesses can come about as 'one off' type accidents in healthy feet, such as treading on a particularly sharp stone, but more commonly they are a 'symptom' of unhealthy, weak feet. So it pays to know what's going on with your horse's feet, to keep/make them healthy & strong. Abscesses draining at the coronet are more likely to be a symptom of lamellar damage/separation or some such. They can also be caused by accident tho - damage to the coronet or kicking a rock.