According to this explanation, the icon first appeared in AD 1420 in Pskov, or rather it appeared miraculously on the Kamenny lake near Pskov, and it was transferred to Pskov the same year.
One copy of it was preserved in Kiev through the Stalin's iconoclasm because a former nun rescued it in her home. It is now back in Kiev's Vvedensky (of the Presentation) Church.
All the copies of it are late, my eye measure gives them 19c. provenance. Here are a couple (I am not saying either of this is the copy presented to the popes).
By the way, the icon seen on the photograph of the two popes praying to, is not it. The icon in question is seen on the first and the last photo, still in the presentation box.
Absolutely beautiful. Mary has a more authentic, middle eastern appearance. Thank you for posting them.