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.
Rationale:
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".
Also, have you identified the relative he was going to? It seems to be "brother i[n] l[aw] Engel" in "New York NY". He might have been from the same town. This is probably a case where it will be easy to read once you know what you're looking at.
– aem Nov 30 '20 at 03:46