How to delete a photo on multiple devices without having to delete it manually everywhere

The only way that I have been able to delete photos that didn't have the trash can icon is with an app called Cleen. Available from the Apple app store. I don't remember if it was free or not. Once downloaded, pull the app up and you will see all your photo albums listed. Click on the album that contains the photos you want to delete, which is probably the All Photos album. When the next screen pops up, you will see the first photo in that album in the center of the screen. Then you will see a little row of blank squares going across the top and bottom. All you have to do is swipe the photo shown up, if you want to keep it, or swipe it down, if you want to trash it. Keep going through the album until you get to the photos that you want to delete. (Just in case you have some photos that you are not sure if you want to keep or delete, you can swipe them to the left and decide on them later) Once you get those photos that you want deleted in the bottom row, hit the trash can. The next screen will show you all the photos that are being deleted and if all is okay, hit Delete at top right. Then a message will pop up saying "Allow Cleen to delete photo from album" and then you click Delete All. It sounds like a lot, but it is really easy. Doesn't take hardly no time. I hope this helps you. This is the only way that I have been able to delete photos when there was no trash can option available.

If you have activated Photostream, why do you use photo sync by iTunes? Just turn it off. And check setting about HDR version of your shots. May be you just see original and HDR version of shot?