I just bought one of these:

Jensen | SMP-xGBL - Digital Audio Player

But upon turning it on, even though the light comes on, the screen is blank. Any ideas?

Can only suggest hitting reset

Look for pinhole and use paperclip to push in to reset the player

If the player does not have a pinhole, remove battery for minute or two and replace to see if this will reset the device

Turn on, hope this works out