The idea that I got is for my kenku character to split phrases and use two or more to make a sentence about what I want to say. Is this possible?
Combine phrases? According to VGtM, they can
"use two or more to make a sentence about what I want to say" — this is how Kenku's "lost voice" curse is described in the Volo's Guide:
Most Kenku use a combination of overheard phrases and sound effects to convey their ideas and thoughts
Split phrases to words and letters? That's unlikely
It seems a phrase is the smallest grammatical unit of speech a Kenku can operate. They use existing phrases (not words) and "variety of noises" to construct their names:
Given that kenku can duplicate any sound, their names are drawn from a staggering variety of noises and phrases.
Hard roleplaying is probably a bad idea in this case
For the sake of speed and clarity, don't try to use only phrases you've heard when communicating with the party. VGtM describes this as distracting:
Roleplaying a Kenku
If you're playing a kenku, constant attempts to mimic noises can come across as confusing or irritating rather than entertaining.
It suggests to describe your character's intentions OOC:
Be clear about your character's intentions unless you're deliberately aiming for inscrutable or mysterious.
Kenku can only repeat sounds they've personally heard. Nothing says they have to repeat the entire sound.
Mimicry: You can mimic sounds you have heard, including voices. A creature that hears the sounds you make can tell they are imitations with a successful Wisdom (Insight) check opposed by your Charisma (Deception) check.
A kenku could imitate the Guard Captain's comments about the city wall patrol and a fisherman's description of a particular catch to explain that "they flopped over the ... wall right through the patrolling guards".
Like any interaction with the rules, the DM has final say. So make sure to ask your DM, as applicable.
No. This is how language-use happens for everyone, and this eliminates the whole curse.
'Hearing sounds, learning meanings, and cutting them up and rearranging them into sentences' is how language skills form in the first place. Before long, your Kenku can hear the few thousand most common words, and 'repeat' them as needed, thus talking like everyone else, with no more impediment than a thick accent.