How do you clear deleted photos?

Yes, as Jim says once the original images are imported into Lightroom, unlike Lightroom Classic, there is no longer any link or reference between the photo in Lightroom and the file on your locally disc. Whatever you do to this disc file  (delete it, move it to another drive) has no effect on the file in Lightroom.

In Lightroom's Preferences you can opt to

store a copy of the photos in the Cloud on your local disc. However, these are mainly meant to provide faster access when working. They are not meant to be used outside of Lightroom and if you delete photos from this location, they will not be deleted from the Cloud. They shouldn't be considered a back up.

Someone else is going to have to give you "real" advice because I am primarily a Lightroom Classic user. But as I understand it, the images on your hard drive are copies of the images in the cloud. The images shown in Lightroom 4.1 are the images in the cloud, the images you can edit, the images that have all the adjustments that you have made using Lightroom. The images that are stored on your hard drive are simply backup images, and they don't reflect any of the work that has been done in Lightroom. If you delete those backup images, as I understand it they are simply gone. But that doesn't impact what is stored in the cloud. With Lightroom 4.1, what is in the cloud is your "official" Lightroom library of images. What is on your hard drive is a backup, and may or may not be the same set of images. And that would depend on how you are managing that set of backups.

I'm a little confused. The images stored on your hard drive are a backup of what is stored in the cloud. The "All Photos" displayed in Lightroom 4.1 are the images that are stored in the cloud, not stored on your hard drive. You have to delete them from within Lightroom 4.1. And when you delete them they go to the deleted folder that is listed in Lightroom, and retained for a period of time in case you change your mind. You can go to that folder and remove them permanently if you choose to do so. If you delete images from your hard drive then you are deleting backup images.