Unleash Incarnation can be used with haste
The haste spell notably limits its own Attack action to being one weapon attack only; however, Unleash Incarnation is not part of the Attack action. We can compare it to the Gloom Stalker Ranger's Stalker's Flurry feature (emphasis mine):
[...] Whenever you take the Attack action, you can make one additional melee attack from the echo's position. [...]
[...] Once on each of your turns when you miss with a weapon attack, you can make another weapon attack as part of the same action.
As we can see, when a feature adds an attack directly to the Attack action, it states this directly. Therefore, Unleash Incarnation is not part of the Attack action and is not limited by haste's "one weapon attack only" clause.
There is also a similar question about haste and Horde Breaker (I will quote its highest-scoring answer):
Horde Breaker says:
Once on each of your turns when you make a weapon attack, you can make another attack with the same weapon against a different creature...
Since haste grants the ability to make one weapon attack as part of an Attack action, it will definitely be a valid trigger for the Horde Breaker ability.
This will work even though haste specifies that the Attack action can be used to take only one weapon attack, because the Horde Breaker attack is not made as part of that Attack action. It just happens to be triggered by that action, so it does not violate that stipulation.
Similarly then, Unleash Incarnation is not part of the Attack action and can be used with haste.
Horde Breaker and Unleash Incarnation use very similar wording and I would therefore treat them the same, that is, that they both can "get around" the restriction from haste.
[...] Whenever you take the Attack action, you can make one additional [...] attack [...]
[...] When you make a weapon attack, you can make another attack [...]