Brit, not a teacher.
I can't think of a verb for this specifically, but there may be one.
Maybe this helps:
To gig = to put on a musical performance (a gig). Not necessarily impromptu though.
gig - definition of gig by the Free Online Dictionary, Thesaurus and Encyclopedia. (Sense 4)
To do a turn = to provide entertainment, usually as part of a larger show, but maybe at a party.
Hope that helps.
Student or Learner