No, a Kenku will never be able to speak fluently
The Kenku were cursed by their previous Master after betraying it. The curse came in three parts (Volos PP 109):
- The wings of the Kenku were withered and fell away, leaving them bound to earth.
- The spark of creativity was torn from their souls.
- Their master took away their voices.
In addition, they can only be Hopeless Plagiarists (Volos PP 110)
Although unable to speak in their own voices, kenku can perfectly mimic any sound they hear...kenku cannot create new sounds and can communicate only by using sounds they have heard. Most kenku use a combination of overheard phrases and sound effects to convey their ideas and thoughts.
The phrase highlighted above further shows that while a Kenku can read and write Common/Auran, they can only communicate with others via Mimicry, and that mimicry is limited (no passing notes in class!)
On page 111 of Volos, there are further directions for roleplaying a Kenku that describe how you this can be done at the table.
The curse given to them by their Master has negated their ability for fluent speech. Their ability to mimic allows them to communicate, but not fluently. Fluent speech would be a removal of the third part of the curse, which has not, and likely will not occur by simply using their current capabilities.
The pursuit of fluency could join their pursuit of flight as a character/plot hook.
Additional insight from Jeremy Crawford
Jeremy Crawford gave some more insight into Kenku:
friend will be playing a kenku in a one-shot. We discussed how mimicry works. Are they more like a parrot, repeating what they've heard? or can they observe a group for a time and make sentences using said voice? how long can they remember sounds?
A kenku is capable of speech. But that speech is a bizarre mix of syllables, words, and phrases in voices the kenku has heard.
More about kenku: they can cleverly piece together voices and sounds they've heard to communicate. They're not less intelligent than other humanoids, but they do have an unusual form of communication. To simplify roleplaying one, think of two-three voices for the character.
These two insights continue to suggest that the Kenku is speaking by combining an amalgam of previous sound captures. This does not equal fluency, but it is more that the more they hear, the more complex sentences they can construct.
Increased efficiency and ability does not equate with speaking fluently, they are only able to mimic a wider range of information.