Newish 1yr battery, lights left on, needs jump start, drive 20 minutes to recharge, turn off, and won't restart, why?

What is your question?

When started with jump engine pulsed with esn and check engine light on with each pulse. After few minutes, pulsing ended, ran smooth. Drove 20 minutes to recharge and wouldn't start after turning off.

What makes this problem better or worse?

Newish 1 yr premium battery, light left on, won’t start, jump start, drive 20 minutes to recharge, turn off and won’t restart.

How long has your 2004 Honda S2000 had this problem?

One day