Photos moved to trash still displayed in Photos App

If you're just right-clicking photos within Finder and selecting Move to Trash, this is only going to affect those particular files. For example, if you're doing that to JPEGs etc in your Pictures folder, you're only moving/trashing those files.

The Photos app (and iPhoto before it) store all imported pictures in a library. You can't delete these manually outside of the app (at least not without showing package contents and causing a whole heap of issues with metadata, etc). Instead you need to delete the photo(s) from within the Photos app.

To do this:

• Within the Photos app select the photos you want to delete. (You can use Shift + click or Command + click to select multiple photos)
• Right-click on any of the selected photos
• You can choose to delete the photos (or even hide them if that's all you really want to do)

NOTE: When you delete a photo it actually places it in the Recently Deleted trash can (appears above your albums). The photos will remain there until you choose to recover them, or until they are automatically deleted after 30 days. You can, however, choose to go in and manually delete them from there anytime prior to the 30 day period.

First, make sure you've actually emptied trash. Next, as @Tetsujin said in the comments, make sure to delete them from Photos (inside the app), can be done on any device and on the next Cloud sync you should see them disappear from iOS and MacOS/OS X devices as well.