1d6 extra damage on each hit, noncumulative
When it hits
The phrase "when it hits" is used at least twice in the PHB, once when describing how the Strength modifier is applied to weapon damage (p. 15, Step 5 sidebar) and once in the spell crusader's mantle.
In both cases (and especially the first) it is clear that "when it hits" means "each and every time it hits, including multiple times in one turn". It certainly does not mean only "the next time it hits", as there are many spells (including most smites) that use just that language.
Since your barbarian gets Elemental Cleaver at 6th level, and already got Extra Attack at 5th, they would indeed be able to apply the damage bonus each time they hit, potentially multiple times per turn even if generated from a single Attack action. Each time you hit with the weapon, you add 1d6 damage on top of the weapon's damage.
You also might think that
it deals an extra 1d6 damage of the chosen type when it hits
could mean that the damage bonus increases by d6 each and every time it hits, so that on the first hit it was +d6, the second +2d6, etc. Such a progression in damage, especially where the bonus overwhelms the base amount, is rare in the 5e rules, and would be clearly indicated as such. In this case the "extra damage" is only 1d6, and nothing indicates that the amount of extra damage increases.
One case where such a progressive increase does occur is the spell delayed blast fireball which says (emphases mine):
A creature takes fire damage equal to the total accumulated damage on a failed save, or half as much damage on a successful one.
The spell's base damage is 12d6. If at the end of your turn the bead has not yet detonated, the damage increases by 1d6.
Here we are told specifically that the damage from the spell 'accumulates' and that it 'increases' so at the end of each turn. If we were to write Elemental Cleaver like this, it would say something like:
(for example) While you wield the infused weapon during your rage, the weapon’s damage type changes to the chosen type, it deals an extra 1d6 damage of the chosen type that increases each time it hits, and it gains the thrown property...
Another example of progressive damage is the falling rule, which says:
At the end of a fall, a creature takes 1d6 bludgeoning damage for every 10 feet it fell, to a maximum of 20d6.
If Elemental Cleaver did progressive damage, it could also be worded something like:
(for example) While you wield the infused weapon during your rage, the weapon’s damage type changes to the chosen type, a creature hit by it takes an extra 1d6 damage of the chosen type for every time the infused weapon has hit, and it gains the thrown property...
Because this additional clarifying language is not present, we can be confident that Elemental Cleaver is a set, not a progressive, addition to damage.