You can yes.
If you use their App to import into OnSong, you'll open a song > Tap the More icon (3 dot icon in upper right hand corner) > Share > Open With... > Tap Add to OnSong -- ((If "Add to OnSong" isn't on the main list there, tap the the more option - tap the green plus sign by Add to OnSong)) > Switch over to the OnSong app > Import > Duplicate (after the first time)...
This can only be done one song at a time.
Just so you are aware.. There is a bug in their app, that it is stripping the name in the metadata section. So when you import a song into OnSong, every new song imported is overriding the last...unless you tap Duplicate and go in and add the metadata information right away.
So if song one is Last Christmas, it'll import into OnSong but you need to edit it to look like the following:
Last Christmas
Wham!
Key: D
Then the next song you import, tap Duplicate and then go in and edit that song. So on and so forth.
According to UG's support team their developers are aware of this and they are working on fixing it. Unfortunately there isn't anything that can be done in the meantime.