Moving photos from Photos app to my iPhone
I would first send the photos on your iPhone back to your Mac using AirDrop. This way you will get ALL iPhone photos on your mac, which you can then import in the Photos app on your Mac. After that you can then sync all photos back to your iPhone with iTunes:


First make a successful backup of your iPhone with iTunes.


Iphone:


Settings -> General -> AirDrop -> Contacts only


Mac:


Open Finder
Click Go in the menu bar and select AirDrop
Allow me to be discoverd by -> Contacts only


Iphone:


Open Photos
Select all photos
Tap the sharing button
Wait untill AirDrop finds your Mac and click it


Mac:


accept the popup window asking you to accept the photos
Import the Photos to the Photos app
Lastly, you can now sync your iPhone with iTunes to transfer all photos/the albums you want to your iPhone. But note that this will erase the media items synced with the previous library (music,podcasts,audiobooks etc.)