Some of my photos were not saved on my iPhone 13 mini

If iCloud Photo has not been updated before you switched to your new iPhone, then your photos, as frazzm737 writes, are probably gone.

You can always see an overview of all your photos in iCloud at:

https://icloud.com - sign in with your Apple ID and look in Photos.

If the photos you are missing are not there, or in Recently Deleted, then they have not been synchronized. Last chance is to check the old phone, but if it has been reset to factory settings, then the pictures are definitely gone.

You have learned a valuable lesson - always back-up your photos and important data to your computer or another device. If those photos do not appear in photo app or Recently Deleted File, they are probably gone for good.