No studio version available, though the live version from
This Left Feels Right: Live is the closest you'll get to to a Jon-only version.
There is another version as a duet with a Russian artist called Alsou which was only played on Russian radio stations. (not included on the album) I haven't heard this version but a few people have said it's better than the Olivia d'Abo version... but that's not really saying much.
EDIT: Here's the Bon Jovi & Alsou version...
It's definitely less offensive than the d'Abo version but still doesn't really do it for me.
"Prayer '94" still pisses over it.