A few days ago I got an email from an Israeli performer looking for the lyrics to this song which she had heard sung by Shifra Lerer (right, with husband Benzion Witler). My correspondent reverse-translated the title as "A rakhmones dayne trern." A shod does mean "a pity" (like "too bad" - if somebody steps in the mud and wrecks their shoes, that's a shod) but it doesn't mean pity/compassion the way rakhmones does.
A shod is also translated as "a waste" and that's more the meaning here.
I pointed the performer to Wolf Krakowsky, specifically to his cd Goyrl (Fate).
The song has also been performed in recent years by Karsten Troyke (A shud dayne trern).
The original song, here performed below by Tadeusz Faliszewski, was Szkoda twoich łez, dziewczyno (1929), translated as "It's No Use Crying, Girl," a tango performed at Morskie Oko, music by Artur Gold, words by Andrzej Włast.
As he often did, Benzion Witler, husband of the original performer Shifra Lerer, wrote Yiddish words to the song.
Here's the sheet music (with transliterated lyrics, translation, and chords) if you want it: