"Kati is looking for banks beside the market."
Translation:Kati bankokat keres a piac mellett.
Hungarian is more "topic-focus-verb-additional" than anything else. Putting the verb at the end is okay in principle. The neat point here is that bankokat doesn't carry an article or anything that makes is special, so it wants to be in the focus instead, i.e. right in front of the verb.
Miskolc, that wouldn't make much sense. Remember that mellett is a postposition, so it modifies the noun that appears immediately in front of it. "A bankok mellett" would mean "next to the banks", and you cannot use both -at and mellett on the same noun.
Since we want to say "beside the market" here, you have to have the construction "a piac mellett" somewhere.