The Life Cleric has 2 separate features which you've conflated here. The first one is Disciple of Life.
Whenever
you
use
a
spell
of
1st
level
or
higher
to
restore
hit
points
to
a
creature,
the
creature
regains
additional
hit
points
equal
to
2
+
the
spell’s
level.
The second one is Blessed Healer.
When
you
cast
a
spell
of
1st
level
or
higher
that
restores
hit
points
to
a
creature
other
than
you,
you
regain
hit
points
equal
to
2
+
the
spell’s
level.
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:
- The first is that using the spell is the same thing as casting it, and therefore only the initial 4d8 + 15 gets the +9 hp from Disciple of Life.
- The second is that you're using the spell to restore hit points every time it restores hit points. In that case, the target would indeed gain 10 hp every turn for the duration.
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 I’m a cleric/druid with the Disciple of Life feature, does the goodberry spell benefit from the feature? Yes. The Disciple of Life feature would make each berry restore 4 hit points, instead of 1, assuming you cast goodberry with a 1st-level spell slot.
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.