prefectural police (Osaka or Kyoto)
Refers specifically to the police force of Osaka or Kyoto Prefecture. Not used for other prefectures; those are called 県警 (kenkei).
The prefectural police (of Osaka/Kyoto) began investigating the case.
The Kyoto Prefectural Police are focusing on tourist safety.
The Osaka Prefectural Police are investigating the case.
I reported a suspicious person to the prefectural police.