This is the 3041st most frequent Polish word.
pobytu
"Stay" or "residence."
Here, 'pobytu' is used in the genitive case to denote the duration of 'stay'.
Długość naszego pobytu w Polsce wynosi tydzień.
The duration of our stay in Poland is one week.
In this sentence, 'pobytu' is used in the genitive case associated with 'during' ('podczas'), implying the period of stay.
Podczas mojego pobytu w szpitalu czytałem wiele książek.
During my stay in the hospital, I read many books.
Here, 'pobytu' exemplifies possession or description, related to 'purposes' ('cele').