What kinds of photos were skipped? As long as your auto-import settings allow what you expect (photos, screenshots, videos) really any of these should be imported provided that the images are not exact duplicates of files that are already in your account (duplicates are skipped).
Also, your device needs to have enough free space to allow import to work. If you are trying to import 5 GB of images, you need at least that much free space on your device so that Lightroom can import the images.
I'll send you a message in a moment asking for a diagnostic log -- if this happened recently enough, maybe the log will help us understand why files were skipped.