Things are a little unclear in the situation you’re trying to describe. Exporting in LR doesn’t delete images. Using Save As in PSE doesn’t delete images. You refer to The Photo and The File in your description but there are at least three different images involved: the original master photo imported into LR, the photo Exported from LR, and the photo created with Save As from PSE. It would also help to know how you’re viewing the images, in Explorer, in PSE’s Open or Save As panel, or in LR. Maybe some of the images are hidden in some of those contexts.
Can you repeat your scenario but using those specific image names, and locations, and use image names that describe which of the three photos you’re talking about.
You might also repeat the procedure but use a folder that isn’t your desktop, maybe one off the root of C:\whatever or maybe your Pictures folder, for the Export from LR and the Save As from PSE.
I originally imported the master image as a CR2 raw from a Canon 6d. When I finished editing it, I exported the modified image as a jpg to my desktop, which has 5 folders and two shortcuts. When I closed LR, I noticed the jpg was no where to be found on my desktop. I looked in all the folders and moved them thinking the pic might be under a folder. Still not visible. I then searched from Cortina (sic?), the Windows 10 search engine. It found the file and when I clicked on it, the Windows picture viewer opened it. I clicked on properties and the viewer reported it as located on the desktop. I closed the viewer, searched again and this time, right clicked on the image and opened it with PSE14. I did a couple edits and did a "Save as" to a different name but also on the desktop. I then closed PSE14. The image I saved from PSE14 was on the desktop, but the exported from LR jpg wasn't. I searched again via Cortina and it appeared. I clicked on properties again. Location was desktop and nothing else like hidden, etc was checked.
As suggested, I opened the master again in LR and exported it to a specific folder on the desktop. That worked fine.
Still a mystery to me.
Each user, and a special All Users or Public profile, have their own Desktop folder. Did LR export to another one of those users' Desktop folder? I am not sure what to tell you since I'm not looking over your shoulder while you're doing the LR Export. I would say check the path in LR Export and make sure it is for your own user not another user.
Good question, however, I'm the only user and always log in as admin. I'll be more aware of the destination folder in the future? When I selected the output folder, the default scroll down "browse" was my desktop. Also, when I checked properties on the same image from PSE, it listed the same destination as the export from LR. A mystery of life with an explanation out there somewhere. I think I'll try a few more exports and see if I get the same results. But, this time, type in the destination rather than just select. Who knows, I might stumble on to something. Thanks for the assistance. TL