Two of the ten powers give you bounty/bond bonuses. Zachary Hudson (President, Federation), and Arissa Lavingy-Duval (Empress, Empire). The others have different bonuses for exploration or trading.
In order to get the bonus:
1. Pledge to either of the two which grant you the combat bonuses (ZH, ALD).
2. Earn at least 100 Merits for them by any means (Fortification, Control, Undermine), and turn them in at a station in a system CONTROLLED by your power. You cannot just turn them in anywhere.
3. Wait until the Thursday cycle.
4. You now have your 20% bonus on bounties and bonds -
- However there is a catch. The bonus only applies in (someone please step in and correct me if this is not accurate) CONTROL systems for that power, and systems EXPLOITED by that power.
For example you cannot pledge to Hudson, and go bounty hunting in Imperial/Alliance space and expect the bonus to apply. You need to stay in the bubbles of systems controlled and exploited by your chosen power to receive the bonus on bounties/bonds.
ALSO - only half of your merits transfer from on cycle to the next, so you will need to earn a few merits each week to maintain the rank (2) necessary to receive the bonus.