I was able to obtain the Petition for Naturalization from the National Archives showing the handwritten city of Asszumki. I have checked the shtetl finder. On my grandmother's Petition she stated he was from Odessa, Russia but I have no proof of this. His mother immigrated from Chezgany, Nizhny Novgorod. They spoke Yiddish. They were not Hungarian.
That appears to be the index card for the naturalisation petition in New Haven Connecticut. The birthplace may well be mis-transcribed by the indexer.
I think you can check the original naturalisation documents on FindMyPast or Ancestry (I haven't used these particular collections before, so I'm uncertain about the exact nature of the contents). I don't have a subscription at home any more, but I can try to check for you next time I'm at the archive.
I think I've identified your ancestor's name (initials M.M.H?) from the dates in the image using the free collection on FamilySearch.
I was able to check the records on Ancestry. They have the original Declaration of Intention, filed in the Superior Court of Fairfield County in Connecticut. It looks like the spelling of Asszumki is correct. It appears twice on the form, first as his place of birth:
and then as his last foreign residence:
I ran a few searches for place names starting with Assz- on JewishGen, and every hit was in Hungary (as suggested by Márton Molnár in the comments), but I still couldn't find an exact match.
One further piece of information from the Declaration of Intention that might help you track him down concerns his arrival in America. He stated that he arrived in New York, from Hamburg, Germany, in 1891. Unfortunately, the name of the ship is given as "unknown".
Just to complicate matters a little more, Ancestry also has Maxwell's draft card. On that he stated that his place of birth was St. Petersburg:
Hope some of this helps.
I think the next things to check:
- the cities, towns and villages names were probably written as the clerk heard it. The clerk does not have any idea what is the correct spelling of the place where application's owner came. And it is normal.
- A lot of people from Belarus and Poland area migrated to the USA. I bet that this man originates also from this region. It is uncommon if he came from, for example, Siberia :-)
- and there is a LOT of villages in Belarus-Poland area. Also their namings were changed or have different spelling. So from provided data it is nearly impossible to identify the birthplace
"Asum" is Estonian for "residential area", "neighborhood". The ending "-ki" could mean plural number number in Russian. I don't believe that my assumption is any truthful though.