If I push a unit into a battlefield while it's damaged and then Relentless Pursuit it to the other one, is it healed from its marked damage?

No, your unit is not healed simply by moving it from one battlefield to another using Relentless Pursuit.

According to the rules and FAQs regarding healing:

1. Healing is restricted: Units only heal at the end of combat (after the Resolution Step) or at the end of the turn (Rule 418.3.a, FAQ #7796, FAQ #7008).
2. Movement is not combat: Moving a unit to a battlefield (even if it causes a showdown) is not the same as combat. A showdown is only considered combat if it proceeds through the Combat Damage Step (Rule 460, FAQ #7796).
3. Relentless Pursuit is an effect: Moving a unit via a spell effect like Relentless Pursuit does not trigger the combat cleanup or any other healing event.

Summary of the interaction:

• If you move a damaged unit to a battlefield, it remains damaged.
• If you then use Relentless Pursuit to move that unit to another battlefield, it remains damaged.
• The unit will only heal if it participates in a combat that reaches the Resolution Step, or when the turn ends.

This is my interpretation based on the rules for healing and combat. No official FAQ exists for this specific interaction, but it follows the established rules for when healing occurs.