If you want to look further on this town, the Hungarian name variants are Nagygejocz, Nagy-geocz etc.

As far as I can tell, birth registers would have been kept in the district seat Ungvar /Uzhgorod/.  Ужгород / Some of the relevant registers have been uploaded to the Wikipedia here https://uk.wikipedia.org/wiki/Єврейське_містечко#Метричні_книги_синагог_до_1895_року (some cyrillic alphabet needed). 

All the best