Life Domain gives you Blessed Healer, which gives additional 2 HP + Spell Level when you cast any healing spell, and the caster gets the benefit of the spell as well.
How does this work with the Regeneration spell?
Regenerate
… The target regains 4d8 + 15 hit points. For the duration of the spell, the target regains 1 hit point at the start of each of its turns (10 hit points each minute).
So does the following happen then?
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Initial Casting = 4d8 + 15 + 2 + 7 (Spell Level) or 4d8 + 24
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Benefactor then gains 1 HP on their turn (10/minute)
Does this mean every turn there is a 10 HP gain per turn (1 + 2 + 7 [spell level])
That would mean a HP gain of 100 HP/minute for an hour right, a sum total of 4d8 +24 plus an additional 600 HP for the hour the spell lasts.
Or the effect only add the HP bonus on the initial casting and nothing happens during the spell duration.
Best Answer
The Life Cleric has 2 separate features which you've conflated here. The first one is Disciple of Life.
The second one is Blessed Healer.
Blessed Healer is simple. You get healed when you cast a spell. So Regenerate will heal you for 9 hp, but only when you cast it.
Disciple of Life is a bit less clear. It is triggered "when you use a spell to restore hit points to a creature". There are 2 interpretations of this:
Unfortunately, there is basically nothing in the rules to help decide which interpretation is correct. "Using" a spell is not common terminology - almost everything in the rules talks about casting a spell, not using it.
However, a similar question was answered in the official Sage Advice Compendium.
If Disciple of Life works every time someone eats a goodberry, it should certainly work every time someone is healed by Regenerate. So yes, Regenerate cast by a Life Cleric will heal 10 hp every turn.