Can somebody recommend a clip-on mic that can be hidden for improving the audio on my videos, or is there a small desktop piece that could be off-camera?
So, this won't meet either of your requirements, however I will say that the Blue Yeti is pretty much the standard microphone for any starting Youtuber. You can always position it to where it's not on camera, and it picks up your voice clearly
Most desktop mics can be positioned off camera and still be pretty clear. Though to be honest, when people are giving new YouTubers a chance at all, they usually just want to see funny stuff. They don't usually have as high standards for new YouTubers, so I wouldn't go spending a ton of money on anything.