If I play Piercing Light and apply its damage to a card in a battlefield, but my opponent reacts and moves the card to their base does it negate the additional damage effect on the card?

No, moving the target to the base does not negate the second part of the effect.

According to RiftJudge FAQ #813 and #6726, the word "then" in Piercing Light is a timing instruction, not a condition that requires the first target to remain valid for the second part of the spell to resolve.

Why this happens:

1. Targeting is locked: When you play Piercing Light, you declare both targets. Once the spell is on the chain, those choices are locked in.
2. Resolution: When Piercing Light resolves, it attempts to deal 2 damage to the first target. If that unit has moved to the base (or is otherwise no longer a legal target), that specific instance of damage fails to occur.
3. Independence: Because the second part of the spell is not dependent on the success of the first part, the spell continues to resolve and will deal 2 damage to the second target you chose.

Summary of the sequence:

• You play Piercing Light and declare both targets.
• Your opponent reacts by moving the first target to their base.
• The chain resolves:
  • The first instance of damage attempts to resolve, but the target is now illegal (it is no longer at a battlefield), so no damage is dealt to it.
  • The second instance of damage resolves, dealing 2 damage to your second chosen target.

This is consistent with Rule 356.3.e.1, which states: "The spell resolves even if some or all of its targets are illegal."