Can I react with retreat as my temporary unit is vanishing at the start of my beginning phase? Would it channel one rune exhausted before readying runes and I would have essentially one more channeled ready rune this way?

Yes, you can react to a Temporary trigger with Retreat. The rune channeled by Retreat will not ready until your next turn's Awaken Phase.

Ruling

You can play Retreat in response to the Temporary triggered ability. When Retreat resolves, it will return the unit to your hand and you will channel 1 rune exhausted. Because this happens during the Beginning Phase (after the Awaken Phase has already concluded), that rune will remain exhausted until your next turn's Awaken Phase.

Sequence of Events

1. Awaken Phase: You ready all your currently exhausted game objects (Rule 315.1.a).
2. Beginning Phase (Beginning Step): The Temporary ability triggers and is placed on the chain (Rule 742.1.b).
3. Reaction: You play Retreat in response to the Temporary trigger.
4. Resolution:
   • Retreat resolves first (LIFO): The unit is returned to your hand, and you channel 1 rune exhausted (Rule 417.2).
   • Temporary trigger resolves: Since the unit is no longer on the board, the instruction to "kill this" cannot be executed and is ignored (Rule 356.3.e.6).

Why the Rune does not Ready

According to Rule 315.1.a, the Awaken Phase—where you ready your game objects—occurs at the very start of your turn, before the Beginning Phase.

Because the Temporary trigger and your Retreat reaction occur during the Beginning Phase (after the Awaken Phase has already passed), the rune you channel enters the board exhausted and will not be readied until the Awaken Phase of your next turn. You do not gain an additional ready rune during the current turn.