If I'm a War Cleric, and I cast Divine Favor on myself and I use two light weapons and they both hit, do I add the extra damage twice since both weapons hit, or just once since both weapons are part of one attack?
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Divine favor adds to all weapon damage rolls; every successful hit with a weapon gets the bonus damage.
That's a bonus to your damage rolls, just like increasing your strength gives you a bonus to damage rolls. It applies to every roll.
Many extra action features targets the attack action, such as Extra Attack, Two Weapons Fighting, Flurry of blows, or Archery Master feat. They do not apply to all attacks, for example they are not triggered when you cast an attack spell or use your reaction to do an opportunity attack.
Many extra damage and defense features targets attack(s). When something says when you hit or miss, it also is in this camp. Bless, Divine Favor, Improved Divine Smite, Sneak Attack, Heavy Armor Master feat, enhancement bonus etc. all applies to individual attacks. They applies to any applicable attacks.
So, your Divine Favor bonus applies to every weapon hit (not every damage roll).