The world famous traditional Abkhazian restaurant at the Sukhum seaside is Nartaa. Though I have to note that for traditional Abkhazian dinner it is usually not preferred by locals. It is usually the flat bread with cheese that this place is famous for, what the Abkhazians call achoaf and coming from Georgian in Russian known widely as loshadki hachapur. It is shaped like a boat.
Abkhazians also have what foreigners call Abkhazian pizza, a closed pizza like bread with cheese, so if you say in Abkhaz language you may be mistaken but if you show with your hand a boat shape they will get it. The next question will be what size you want. You have small [malinki; 0,5], middle [srednii; 1] and big [bolshoi; 1,5]. 1 is good if you are having only that or generally eat a lot, but if you will order some extras go for small.
The best thing about Nartaa is its central place and the sea view. However, don’t go there when you are desperate to eat something immediately, as the service is very very slow.
For a drink with hachapur, i would recommend to try atarhun [tarragon] or another local Abkhazian lemonade.
Tea is usually an option for winter but the waitresses can look like you are insulting them any season if you order one. We also made the mistake of asking for coffee for the waiting period and icecream as desert and they all arrived with the main food and we could only laugh at the situation. So dont try to be smart ordering everything at once.
If you have only one chance to have traditional food in Abkhazia you can still order some traditional food which will is detailed in another post.