If a player has no valid targets for a bird power (such as no cards in hand to tuck or no food to cache), they must skip that power entirely.

This is supported by the following rule:

• "Bird powers are always optional. For example, if you do not want to spend a card by tucking it under a flocking bird, you do not have to do so."
  (page 10)

This means you are never required to activate a bird power, and if you cannot fulfill its requirements (e.g., you have nothing to tuck or cache), you simply do not use it. There is no mention in the provided context of partial effects or alternative outcomes if you cannot fully resolve a power.

Source: page 10

Source: Core Rulebook (pp. 1, 10, 7, 8, 9)