The question touches a grey area.
Because the answer primarily depends on what one actually means by the word "speaking". So, a ultimate DM ruling is paramount here.
TL;DR: One can "speak" in his mind, "hearing" his own voice in his head, without actually moving his mouth of making any sound, and thus telepathy counts as "speaking".
The relevant Raw texts:
Kenku racial curse flavor text:
Finally, to ensure that the kenku could never divulge any secrets,
their master took away their voices.
Kenku trait:
Languages. You can read and write Common and Auran, but you can speak
only by using your Mimicry trait.
Warlock Awakened Mind:
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.
Emphasis mine.
As a DM, I see two kinds of telepathy:
Deep Connection Telepathy: The famous "mind to mind" of Spock from Star Trek: Conversing directly in images, feelings, memories, raw concepts, etc. Not by "talking aloud in your mind"! This telepathy type bypasses all language barriers. It is neither" speaking'" nor "writing". However, it can come with it's own set of interesting roleplaying communication challenges. Only mostly limited to some psionic (sorry: mystic now!) powers in D&D.
Classic Telepathy: The much more common form of Telepathy works much like picking up a cell phone and sending a text message. You are "speaking", but only in the sense that you are doing it only in your mind. Similarly, the recipient "hears" what you say, but only in the sense that he hears it "in his head", and overall it feels like it is some mind-voice coming from nowhere.
Usually with Classic Telepathy you don't need to share a language with the target, but the target must know at least one language, and while not written down specifically, it is very strongly implied that the sender is just using one of the language that he knows, too, to talk from his mind alone. It is like typing a text in English and sending it to a French speaking person, and the magic does the translation automatically: Both the sender and the recipient are still using a language, just not necessarily he same one.
Classic Telepathy thus seem to require the sender to "speak a language from his mind" and is a form of "speaking". Mentally, this is the same as speaking aloud, except you don't physically move your mouth, tongue, or vocal chords.
Except for a few mystics (psionics) powers, which are closer to how Deep Connection Telepathy works, Classic Telepathy is the "normal" form of telepathy in D&D, and is also how most DMs and most source-materials (including non-D&D-related sources such as novels and fantasy movies) will depict telepathy (at least, most of the time).
So we can say that Awakened Mind will not bypass the curse.
Now, the deity removing the voice, also means the inner voice, too. If the divine curse was not a mental aspect, but a physical one, meaning if it only affected a kenku's ability to use his vocal chords, then even Mimicry would not work, because the kenku would be a mute. So the curse affects the mind, not the muscles, and thus affects language capabilities directly.
Of course, the DM is perfectly in his right to assume that only the "outside" voice was removed, not his "inner voice", but since RAW doesn't make the distinction, and it is fluff text, I assume the fluff text refers to all types of voices, outer voice and inner voice, and use more the mechanical description than the fluff.
Speaking doesn't necessarily have to be only in the form "talking aloud using your mouth tongue and vocal chords". A person can perfectly "hear" his own voice "in his own mind" when "talking aloud but only in one's own mind".
So, any form of "speaking" counts for the curse. However, RAW says that written language is not affected. But, as per RAW, having an exception for some way to communicate, such as by using written language, does not mean that suddenly every other way of "speaking" (communicating) also gets to be an exception too.
So the question becomes: Is Classic Telepathy "closer" to speaking, or "closer" to writing? While this can be debated, as a DM I tend to apply the most obvious and common sense "Occam's Razor" answer: When you use telepathy, the recipient "hears you" in his own mind. So, you are "talking" to him, in your own mind. This is pretty much some kind of verbal communication, just done along another "bandwidth channel": mental waves instead of sound waves.
Awakened Mind is a typical D&D way to do telepathy, thus of the "Classic Telepathy" type. It is thus a way of speaking "directly with your mind". And thus, the kenku curse applies.
Other considerations.
A DM should consider the side effects of letting any Telepathy bypass the kenku curse, especially since Awakened Mind is at will. thus, very rare will be situations where the warlock PC couldn't just use telepathy to speak to other creatures, instead of just speaking "normally". Basically, that DM would end up with a player not wanting to be affected by his racial penalty.
If the campaign is all about not bothering with roleplaying fluff and just getting on to the next action sequence as soon and as painlessly as possible, then ok, no problem, a DM can allow it. After all, as per DMG page 4:
And as a referee, the DM interprets the rules and decides when to
abide by them and when to change them.
But, by default "RAW", I think telepathy is a way to "speak with your mind", and thus the curse applies.
