If you do not want the images you have in your Lightroom Mobile to get synced with your desktop Lightroom either you can turn of the sync process in either of the application or remove the images from Lightroom Mobile.
If I do that, then the collections on the desktop that I want to sync with won't sync, correct?
I don't want LR Mobile to shove photos/collections to LR Desktop that I didn't ask it to sync. On the desktop side, I have to specifically choose which collections will sync to LR Mobile, so I want that ability in LR Mobile
Yes that can be only done in the Lightroom desktop application to sync specific collection over the mobile version.
I would suggest you to raise a feature request to have such an option in Lightroom mobile as well at http://feedback.photoshop.com/