I'm holding a battlefield with a temporary Sprite Token and I have a hidden Keeper of Masks. Let's say I react with my Keeper of Masks to the temporary trigger of my Sprite and I spawn 3 new temporary units on the battlefield. Would they be cleaned up because I'm still in the Beginning Phase, or would they stay for my turn?

To answer your question: The new units will stay for your turn.

According to the rules and the mechanics of the Beginning Phase, they will not be cleaned up by the "Temporary" trigger that is currently resolving.

Reasoning

1. Trigger Timing: The "Temporary" keyword is a triggered ability that activates at the start of your Beginning Phase, before scoring (Rule 816.1.b).
2. Chain Resolution: When you react to the Sprite's "Temporary" trigger by flipping your hidden Keeper of Masks, the Keeper of Masks enters the battlefield and its ability is placed on the chain.
3. LIFO Resolution: Because the chain resolves in Last-In, First-Out (LIFO) order, the Keeper of Masks' ability (which creates the Reflection tokens) resolves before the original Sprite's "Temporary" trigger finishes resolving.
4. Trigger Window: The "Temporary" trigger check for the Beginning Phase happens only once per phase. Because your new units enter the battlefield after the initial trigger check for the phase has already occurred, they are not subject to the "Temporary" kill trigger until the next time your Beginning Phase starts (Rule 340, RiftJudge FAQ #340).

Sequence of Events

1. Beginning Phase starts: The "Temporary" trigger for your Sprite token is placed on the chain.
2. Reaction: You play your hidden Keeper of Masks in response to the Sprite's trigger.
3. Resolution:
   • Keeper of Masks enters the battlefield.
   • Keeper of Masks' ability triggers and is placed on the chain.
   • The Keeper of Masks' ability resolves, creating the Reflection tokens.
   • The original Sprite's "Temporary" trigger resolves, and the Sprite token is killed.
4. Result: The Keeper of Masks and the new Reflection tokens remain on the battlefield. Since the Beginning Phase trigger window has already passed, they will not be killed by the "Temporary" keyword until your next turn's Beginning Phase.

Note: While the Reflection tokens created by Keeper of Masks do not inherently have the "Temporary" keyword (Rule 184.6), they are created by the Keeper of Masks' ability. Even if they were to gain the "Temporary" keyword through another effect, they would still survive this turn for the same reason: they entered the battlefield after the Beginning Phase trigger check occurred.

Note: I could not fully verify this ruling against the retrieved rules.