I’m quite certain this is not the correct translation.
“Chata” typically means a house in the nature, outside a town or village.
The word we seek is probably something with “záhradka” … within the borders of a town, that can be rented for long periods, intended for growing vegetables.
Hmm I've been to chaty like you described - in the countryside or a village out in south moravia, south bohemia and vysočina. But I've also been to garden plots near to a city with a little building that has a kitchen and room to chill/sleep in (these little guys: https://en.mapy.cz/zakladni?x=16.6371848&y=49.2350039&z=20&m... - maybe hard to see, street view didn't go here :) ) which were also described as a "chata" to me. There are definitely little bare sheds in some of these plots, but in the context we were discussing I feel like "chata" fits well and is analogous to "dacha" from the comment I replied to. Though I've heard of some grander "dacha", better equipped with more amenities that might be referred to as "chalupa" here.
The word we seek is probably something with “záhradka” … within the borders of a town, that can be rented for long periods, intended for growing vegetables.