This is the 4888th most frequent Russian word.
казахстанской
"Of Kazakhstan" or "Kazakhstani" (adjective).
In this sentence, 'казахстанской' is in the genitive case indicating possession or relation to Kazakhstan and modifies 'певицы' to specify an attribute of her being from Kazakhstan.
Я слушал концерт казахстанской певицы.
I listened to the concert of a Kazakhstani singer.
Here, 'казахстанской' is in the prepositional case modifying 'столице' to describe the feature of the capital, showing that it is located in Kazakhstan.
Мы побывали в казахстанской столице.
We visited the Kazakhstani capital.
In this instance, 'казахстанской' is in the genitive case associated with 'кухни' to attribute the cuisine's origin to Kazakhstan.