While the rules don't specifically state that you can break up your movement an unlimited number of times, that certainly seems to be the case.
You can break up your movement on your turn, using
some of your speed before and after your action. For
example, if you have a speed of 30 feet, you can move
10 feet, take your action, and then move 20 feet.
If you take an action that includes more than one weapon
attack, you can break up your movement even further by
moving between those attacks. For example, a fighter
who can make two attacks with the Extra Attack feature
and who has a speed of 25 feet could move 10 feet, make
an attack, move 15 feet, and then attack again.
Since high-level fighters have up to 4 attacks with Extra Attack, this means that you can at least break up your movement into 5 sections. There's no reason why this would be a limit, however - it seems safe to assume that you can break up your movement as much as you please.
In any case, the only thing that it isn't specifically stated that you can break up your movement with is your bonus action. If your DM is an absolute stickler for "it doesn't say you can", then this is the point that you might not be able to win.
The fun way to do this, as well as the obvious interpretation of the rules, would be to have them imitate phrases perfectly, sounding like the original speaker, but using them in approximately the sense they originally heard them. So their voices can change completely from sentence to sentence - one being a dwarf they heard speaking yesterday, the next sentence being a small elf child they heard ten years ago, and the third being something you said to them three months ago.
They understand what phrases and sentences mean, but if they can't break down phrases into words and use the words individually, they're still fairly limited by their curse. They would sound a bit like Dilbert's pointy-haired boss, who tends to come out with phrases and clichés that belong in a subtly different conversation.
This would be an interesting challenge for a player who was usually verbally adroit. Precedents for even more extreme versions exist in the Librarian from Terry Pratchett's Discworld series who manages to get by with just one word, "Oook", and Mr Punch in the classic Punch & Judy show, most of whose lines are "That's the way to do it!"
The main problem with playing a Kenku this way would be avoiding humour in serious moments of the game.
Best Answer
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:
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:
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:
It suggests to describe your character's intentions OOC: