Can you return a card to your hand with buyback and also apply a cost reducer to lower the total cost?

Buyback is an optional additional cost you may choose to pay when casting a spell (CR 702.27a). If you choose to pay it, the buyback cost is added to the spell's total cost before any cost increases or reductions are applied (CR 601.2f).

Cost reducers — such as effects that say 'spells cost {X} less to cast' — apply during the final cost calculation step (CR 601.2f, 601.2h). Crucially, these reducers apply to the total cost of the spell, which already includes any additional costs like buyback. So a reducer can effectively lower the combined base + buyback cost.

However, cost reducers cannot reduce a spell's total mana cost below {0} (CR 601.2h). The mana component won't go below zero, though any additional costs with non-mana components (like sacrificing permanents) must still be paid in full.

Concrete example: You cast Reclaim the Forgotten (hypothetical: {2}, buyback {3}). Normally the total is {5}. If you have an effect that makes spells cost {2} less, you pay only {3} total while still returning the card to your hand via buyback.

So yes — choosing buyback and benefiting from a cost reducer are fully compatible. Just declare buyback during the casting process (CR 601.2b), then apply your reducer to the combined total before paying.