How do I delete specific photos from my photo-stream on the Mac or iPhone?
In recent (2017) versions of Photos.app, individual images can be selected and deleted from "My Photo Stream". These photos may also be removed from the local library on the device (depending on exactly how they were deleted: delete key; menu items; contextual menu; etc.), but not on other devices that have imported the images into their local library (which happens automatically with Photos.app on the Mac). Photos that were removed from the local library can be recovered from the "Recently Deleted" folder on the device, and are not added back to "My Photo Stream". Note that the photos visible in "My Photo Stream" can differ between different devices, perhaps due to time lag between local caches and server updates.

In earlier software## Heading ##, it seems that the only way to do this is to turn off Photo Stream, delete the photos from your devices, and turn Photo Stream back on.

From the relevant Apple support page:

To Reset your Photo Stream, follow these instructions:

    From a computer browser, go to icloud.com and sign in.
    Click on your name on the iCloud home page (this is the page that shows all the app icons). If you do not see your name, click the cloud icon in the top left of the page.
    Click Advanced.
    Here you will see the Reset Photo Stream button. Clicking this button will remove all your Photo Stream photos from iCloud. Once removed, they cannot be recovered, but you can re-upload any photos you'd like to restore to your Photo Stream from a Mac or PC.

After resetting your Photo Stream, you can also remove any photos that have been downloaded to your device or computers. To do this, follow the directions below:

    On your iOS device, you can remove photos from the Photo Stream view by turning off Photo Stream and then optionally turning it back on. You can do this at Settings &gt; iCloud &gt; Photo Stream.
    If you added any Photo Stream photos to other albums or the Camera Roll on your iOS device, you can locate and delete these photos manually, if desired.
    
    On your Mac, you can clear cached photos from the Photo Stream view in iPhoto or Aperture by turning off Photo Stream and then optionally turning it back on. You can do this in each application by visiting Preferences &gt; Photo Stream.
    If you had Automatic Import enabled in your Photo Stream settings, your Photo Stream photos were also imported to your iPhoto or Aperture Library, and will remain after you disable Photo Stream. If you would like to delete any of these photos from your library, you can to do so manually any time you like.
    
    On your PC, you can remove Photo Stream photos by deleting them from your designated Photo Stream Download Folder. By default, this folder is located at C:\Users\Apple\Pictures\Photo Stream\My Photo Stream.