It's a bit of a guess, but I'm bold and adding this as answer rather than a comment. I think it is Kysak (Hungarian: Sároskőszeg) half-way between Prešov and Košice.
The first writer had a tendency to move letters that near together so that they might appear as one. You see this for example with "Austria" on line 10 where s and t nearly are one. In the same way the first letter of the town is a "S." (first I thought it might be "St."), then the next letter is a "K" (quite apparent at least to me) and then we have again a letter that near that it nearly vanishes. It is a "o" and the accent is shifted to the right. That is also why the "s" is cut off. Then I think a "e" follows and the last one is clearly a "k" again.
That makes "S. Kösek" where the writer might have misheard "kőszeg". On the other page we have a different handwriting and there I read "Kosz." So this writer heard better. Not sure why the accent got missing though (ő [note the acute accent] is pronounced longer so no chance to mishear that for an o). Maybe we see it in the sloppy connecting line between the "o" and the "s".