Genealogy / Hetman Stanislaw Jan Jablonowski and Marianna Kazanowska [NEW]

Hi, I'm wondering if anyone knows more about the issues of this noble couple. According to various genealogy sites - they had a son born in Krakow around 1678. He later moved to Sweden (around 1709) with Stanislaw Leszczynski and stayed there ever after. He is known by swedish records as Johan Joseph Jablonski - not Jablonowski.

Any chance this birth in Krakow was ever recorded in polish archives? I havn't seen any proof of that yet. Was this son ever concidered a "true" Jablonowski? And if he wasn't - howcome the genealogy sites mention his parents to be as in the headline of this text?

Thank you so much for reading and have a good day/
Marten