die Verwicklung
involvement
Participation in, or being drawn into, a complicated, unpleasant, or criminal matter.
Seine Verwicklung in den Finanzskandal kostete ihn seine Karriere.
His involvement in the financial scandal cost him his career.

involvement / complication
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involvement
Participation in, or being drawn into, a complicated, unpleasant, or criminal matter.
Seine Verwicklung in den Finanzskandal kostete ihn seine Karriere.
His involvement in the financial scandal cost him his career.

The noun 'Verwicklung' comes from the verb 'verwickeln', which means 'to confuse' or 'to draw into something'. The root 'wickeln' (to wrap) creates the image of being 'wrapped up' in a complicated situation.
Er wollte sich nicht in die Streitigkeiten seiner Kollegen verwickeln lassen.
He did not want to let himself get drawn into his colleagues' disputes.
When the word appears in the singular and often with a possessive pronoun ('seine Verwicklung' — his involvement), it usually refers to personal involvement. In the plural ('die Verwicklungen' — the complications), it more often describes the complicated circumstances themselves.
Seine Verwicklung in die Affäre wurde nie bewiesen.
His involvement in the affair was never proven.
To express what someone is involved in, the preposition 'in' is used with the accusative. Typical phrases are 'Verwicklung in einen Skandal' (involvement in a scandal) or 'in eine Straftat' (in a criminal offense).
Die Polizei untersucht die Verwicklung des Managers in den Betrugsfall.
The police are investigating the manager's involvement in the fraud case.
When 'Verwicklungen' describes complications, it is often combined with adjectives such as 'politisch' (political), 'diplomatisch' (diplomatic), or 'rechtlich' (legal). Another common phrase is 'es kam zu Verwicklungen' (complications arose).
Der Grenzstreit führte zu ernsthaften diplomatischen Verwicklungen.
The border dispute led to serious diplomatic complications.
difficulty
'die Schwierigkeit' and 'die Verwicklung' can both describe a complicated situation. 'Schwierigkeit' is more general and means a problem or obstacle. 'Verwicklung' emphasizes more strongly that people or events have become caught up in a complicated, often unpleasant matter. The word is especially suitable when a situation becomes confusing because of relationships, secrets, mistakes, or multiple events. For a normal problem, you usually say 'Schwierigkeit'. When someone is drawn into a complicated affair, 'Verwicklung' fits better.

Durch seine Verwicklung in den Streit konnte Paul nicht mehr neutral bleiben.
Because of his involvement in the argument, Paul could no longer remain neutral.
Vor dem Termin gibt es noch eine Schwierigkeit mit den Unterlagen.
Before the appointment, there is still one difficulty with the documents.