A Kenku can only speak by mimicry, so can a Kenku fulfill the verbal portion of spells of his class? I would assume that he could only cast a verbal spell if he has heard the spell be cast by another caster of his same class. Also, the other caster would have to have cast the spell in a way that the Kenku knew what was spoken and why.
A kenku should generally be able to produce the required sounds
Taking a look at what the verbal spell component consists of, we have this (my emphasis):
Most spells require the chanting of mystic words. The words themselves aren't the source of the spell's power; rather, the particular combination of sounds, with specific pitch and resonance, sets the threads of magic in motion.
As long as the kenku can cobble together sounds with the required pitches and resonances, it doesn't really matter how they do it. The sounds don't even really need to consist of speech. Even a kenku with a very limited vocabulary will have a wide repertoire of other sounds that they could mimic to fulfill the verbal components, such as birdsong, animal calls, the sounds of rushing water, etc. In short, unless the kenku was raised from birth alone in a soundproof room, they should be able to produce a sufficient variety of sounds to provide the verbal components of spells.
So, rather than making this a limitation, use this as an opportunity to add some unique flavor to your kenku spellcaster by indicating what kind of unique sounds they are using to provide the verbal component to each spell - perhaps the sound of a rushing river to cast tidal wave, the crackling of a bonfire to cast burning hands, or a bird's mating call to cast charm person. Or work it into your backstory by creating (in collaboration with your DM) an NPC mentor who taught them all the basic phonemes they would need to combine in order to cast verbal spells.
Yes, they can cast spells with verbal components.
See this unofficial tweet by Jeremy Crawford from December 2016 addressing this exact question:
if kenku can't speak, does that mean no spells with verbal components (i.e. almost all of them)? Mimicry sufficient?
Kenku speak by mimicking sounds they've heard. They're not prevented from uttering verbal components.
This might seem confusing at first. However, disallowing Kenku from casting spells (or imposing massive restrictions on doing so) would basically render Kenku unuseable as casters. Hence, for balancing reasons, they are allowed use verbal components.
Crawford justifies this ruling by using the phrasing "uttering verbal components", which suggests that verbal components are not real words. Either way, verbal components (and somatic components, for that matter) are notoriously loosely defined, aside from being audible.
Despite Crawford's tweets being no longer official, I recommend following his advice.
There are no newer / more official rulings on this topic as far as I can tell. However, Crawford's tweets still give an indication of the design intent.
Regardless of this, it would be horribly restrictive for players wanting to play a Kenku if they then couldn't play any caster class (which doesn't leave a whole lot of options tbh). 5e doesn't restrict players like this at all, aside from the basic racial traits - which only slightly encourage certain combinations or discourage others, but don't prevent you from playing, for example, a Tiefling barbarian, despite the racial stats not synergizing at all with the class. This is in contrast to earlier editions (especially pre-3rd afaik), which imposed severe restrictions on your character choices depending on which class or race you wanted.
Thus, I strongly encourage any DM to not disallow Kenku spellcasters, especially not after the campaign has started - otherwise, a player hoping to eventually subclass his/her currently still low-level Kenku fighter into an Eldritch Knight would suddenly have his/her character concept ruined due to a lack of information upfront.
I'm personally certainly also not a fan of making it more difficult for Kenku casters to learn new spells, but that's probably somewhat personal preference.