Can Vex Cheerless reduce the cost of paying deflect?

Yes, Vex, Cheerless can reduce the cost of Deflect.

According to RiftJudge FAQ #9237 and FAQ #8596, the interaction works because of the order of cost calculation:

1. Deflect is a mandatory additional cost (Rule 809.1.d) that is added to the spell's total cost.
2. Vex, Cheerless applies a discount to the total cost of the spell.
3. Because the Deflect cost is included in the total cost before discounts are applied, Vex, Cheerless's discount is applied to that total, effectively offsetting the Deflect cost.

Key Nuances:

• Minimum Energy: While Vex, Cheerless has a "minimum 1 Energy" restriction, this applies only to the spell's base cost, not the total cost including additional costs like Deflect. Therefore, Vex, Cheerless can reduce the total cost (including Deflect) below what would otherwise be the minimum.
• Order of Operations: Per Rule 356.4, discounts to the total cost are applied after all additional costs (like Deflect) have been summed.