Ah, good. I too had a stub on this, may as well do as filler on background:
For anyone not at all classically oriented, a cadenza may be likened to a plugin used within another program. Mainly – bluntly – a device to dazzle audiences, thus also bribe soloists into performing one’s work, it has blossomed through centuries to a repertoire of its own.
Typically, exemplified here with Hillary Hahn’s performance of Mozart’s Violin Concerto 3 with a cadenza of her own, it is set just before the home stretch of a concerto movement (@6:44), at a point where the full orchestra has worked itself up to its highest-charge chord (@7:16, a 6/4-chord) cut off in sudden air = ‘[Insert cadenza here]’. In the composer’s score this may be written out in extenso (which is perfectly OK for a soloist to subsitute for any whim of his/her own, provided aesthetics match), just cued, or not even indicated but understood by convention. The soloist now picks up from the dynamic collapse and, from a mere few measures to several minutes completely on his/her own (sometimes with a little help from friends, such as Beethoven’s solo piano +timpani in his own arrangement of the violin concerto, nowadays as a rule arranged back in the violin version), rebuilds charge to the same point as where the orchestra fell silent, and at a cue of familiarity (@8:46, a typical dominant chord trill), they join again right at the home gates now open for a final straight.
Initially the cadenza was improvised on the spot. Later growing to balance that spot to the unified workmanship of its native surrounding, integrally thematic, soon pre-planned, -pondered … and -penned: composed. The thing is, the cadenza is very often not written by the composer of its parent concerto, is therefore properly credited on classical releases.
The Wikipedia article, under ‘Composed cadenzas’, is definitely misleading in “Joseph Joachim wrote the cadenza for Brahms’s Violin Concerto.” The Joachim cadenza is certainly the most often performed, but one among others, as popular concertante works tend to have cadenzas amassed over time: Here Ricci’s release of Brahms’ violin concerto, with a selection of 16 cadenzas to choose from. This is of some importance, since a somewhat initiated listener will from time to time want to find a recording of this or that concerto with a certain cadenza, or just the cadenza itself; and the constellation (composer-)work-cadenza-soloist may even be what makes a listener like this or that recording.
A cadenza may also quote, or even be based upon, a work completely different from its parent work (further complicating credit). A serious enough instance is this recording: [Mozart’s] Piano concerto 25 I. Allegro maestoso, where András Schiff in his own cadenza (12:52—14:40), for dynamic climax, unexpectedly but in true tradition, quotes [SPOILER](@ 13:55) from Mozart’s Figaro the instrumental march theme interpunctuating-concluding the aria “Non più andrai, farfallone amoroso”[/SPOILER] to great effect.
Schnittke’s first cadenza to Beethoven’s violin concerto deviates further in both idiom and material. Here is Gabriel Bolkosky performing it (cadenza cue @2:45—7:45), try to name the quotes!
I cannot find a Documentation page listing the Work Types available from the Add Work drop-down menu, but among listed ones I certainly find cadenza somewhat more missing than others, because of the above “what makes a listener like this or that recording” and want to (be able to search to) find it, although baked into the precedent parent Concerto Work Type, and . I think it should be added to that list (along with some others, f.i. antiphon, canon, fantasia, fugue, intermezzo, melodrama, offertorium, prelude, psalm, toccata, …).