Move, not copy, photos on an IPad

When you create folders on the iPad and "move" photos, you are not actually moving them. They remain in the Recents, and you are just creating a shortcut in the new album that you are creating. If you were to delete the photo from one of those albums, it will delete in both. The only way to remove a photo from the Recents and put it into a folder without it remaining in the Recents is to import photos from the iPad to your computer, and then sync them back to the iPad with iTunes. At that point, the only way to delete them from the iPad is to change the sync preferences in iTunes. Transfer photos and videos from your iPhone, iPad, or iPod touch - Apple Support and this, Keep your photos safely stored and up to date on all of your devices - Apple Support

Not aware of an app that would allow you to move photos and delete them from the Recents. Albums created as you are doing are for temporary access to separate things. I would just keep them as you have them, and worry about taking them out of Recents at a later time when you have it. You can also review using iCloud photos in the first link I provided.