If I have a spell cost reduction (like empowered Applied Researchers), do spells that originally costed 4 energy still qualify for Jhin legend's ability to be banished, or do I have to actually spend the 4 energy?

No — you have to actually spend 4 or more energy.

Virtuoso's ability reads "When you play a spell, if you spent [4] or more, you may banish it." That condition checks the energy you actually paid during finalization (Rule 357), not the printed cost. With empowered Applied Researchers reducing your spells' Energy cost by 1 (minimum 1), a 4-cost spell only costs you 3, so you spent 3 — Virtuoso doesn't trigger and you can't banish it.

This is the opposite of effects that reference a spell's cost (like Lux or Defy), which use the printed cost regardless of discounts (Rule 206). "Spent" is specifically about real payment — which is also why paying a Repeat cost counts toward Jhin's 4-energy threshold, and why Hard Bargain payments don't (FAQ #10012, #10606).

So: 4-cost spell under Applied Researchers → spends 3 → no banish. A 5-cost spell reduced to 4 → spends 4 → banish works.