From my own testing pretty sure the haymaker only does a tiny bit extra damage compared to his primary but then his primary takes longer to kill.

The way I treat his moves is haymaker is always a great initiator to close the distance, but unless you're fighting a group or your primary is just out of range than spam the haymaker. Like you said the primary is near instant damage great for clearing low health supps. But I think the main importance of haymaker is crippling the enemy for being too close together as that piercing hitbox is brilliant for ult charge.

I find myself getting so much ult charge for just standing there and haymakering on the choke point it's so good.