The troublesome prince is often the person a court cannot quite control. They may ignore etiquette, provoke the wrong people or treat serious situations with an infuriating lack of concern. Their behaviour can be funny, but it should also tell readers something about the character.
A prince who causes trouble may be bored by the restrictions of royal life, angry about expectations or determined to keep others from seeing a more vulnerable side. Sometimes mischief is a form of rebellion. Sometimes it is simply a bad habit that creates consequences.
The trope becomes more interesting when the character is not excused from responsibility. Charm does not erase the effects of their choices. A strong character arc allows the prince to recognise when their behaviour hurts others and decide whether they are willing to change.
Romance can expose the difference between being entertaining and being considerate. A character may enjoy teasing someone but eventually need to learn how to listen.
Yi Yinxiang in Assigned to the Ruined Prince is confined to the Eastern Courtyard for behaviour the court prefers not to describe. His new watcher, Gu Yanzhi, expects a difficult assignment. Their relationship begins with irritation, but the story has room to reveal what lies beneath the prince's troublesome reputation.