Can I get the play effect of something that has an additional cost when it was played with Temporal Breach?

Yes — the Play Effect still triggers, but the additional cost is still on you.

Temporal Breach plays the unit back normally, so "When you play me…" triggers as usual once it enters the board — Play Effects are separate from costs (FAQ #2608). What "ignoring its cost" does is zero out the base Energy/Power cost (Rule 356.1.b.1) — it does not ignore additional costs. Mandatory ones (e.g., "As an additional cost to play me, kill a friendly unit") must still be paid, and optional ones like Accelerate you may pay or skip (Rule 356.1.b.3, FAQ #11822).

If you can't pay a mandatory additional cost, the play-back instruction simply doesn't execute, so the unit never enters and you get no Play Effect. But as long as you pay it, the trigger fires normally.