@NautArch post from 1/2017 states with references:
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.
Not from @NautArch post below
In reference to the Telepathy question for a warlock:
- Officially Awakened mind is 1 way communication only
- Awakened mind would allow you to communicate telepathically with any creature you can see within 30 feet of you since there are no need for shared languages at all and it doesn't refer to this as speech.
Awakened Mind PHB pg 110
Starting at 1st level, your alien knowledge gives you the ability to touch the minds of other creatures. You can communicate telepathically with any creature you can see within 30 feet of you. You don't need to share a language with the creature for it to understand your telepathic utterances, but the creature must be able to understand at least one language.
Since the spell describes them as utterances, not speech, your DM should not have any issue with a Kenku using it.