Wiktionary
vb. (context idiomatic English) To have a tear released, to cry (due to sadness)
Wikipedia
"Shed a Tear" was a single by Wet Wet Wet recorded especially for their first greatest-hits album, End of Part One. It was released on 25 October 1993 and reached #22 in the UK Singles Chart.
Title lyric: I find it in my heart to say, 'I'm gonna shed a tear for you today.'
Usage examples of "shed a tear".
He wondered whether Dickon would shed a tear for his brother who died in the snow, somewhere off beyond the edge of the world.
Sansa would have sighed and shed a tear for true love, but Arya just thought it was stupid.
If she had cracked up without my help, I wouldn't have shed a tear -- but to let her do something too dangerous while she was my pupil.
Alyssa Arryn had seen her husband, her brothers, and all her children slain, and yet in life she had never shed a tear.
Not that Knute would shed a tear if that happened, but if Bear somehow came to his senses and made good his escape, Knute wanted to appear convincing in his role as loyal and concerned pilot.