A Verlegenheit is a difficulty one may find oneself in, tinged with a whiff of embarrassment, and a Lösung is a solution. In combination, they indicate a solution that is less than ideal, the result of necessity rather than choice—there simply was no better alternative available. The so-called fiscal cliff deal we were treated to two days ago looks very much like a Verlegenheitslösung to me—nobody is happy with it, but something better was not doable, and everybody has some 'xplaining to do to their respective constituents.
In Germany, I find the term used particularly often in connection with personnel decisions, like when a coach names someone to a team only because the usual starter or substitute is injured. The whiff of embarrassment comes into play because the implication is that the team does not have enough depth to field a better replacement. It's this whiff that distinguishes a Verlegenheitslösung from what's called in English a stop-gap solution. In addition, it may not (temporarily) stop a gap: A less-than qualified cabinet member, who was a Verlegenheitslösung when appointed, may last for the entire term of a government.
"Bare-handed speech synthesis"
10 hours ago