Future -- Child Support War With Baby Mama #2

Rapper Future stepped up to the plate and took responsibility for ANOTHER kid ... problem is ... now he's battling the baby mama in court over child support.

TMZ broke the story ... a woman named Brittni Mealy slapped Future with a paternity lawsuit back in Feb, claiming he was the father of her son. But a month later she dismissed the suit, without explanation.

Then on June 8th, Future filed docs, admitting he knocked Brittni up and asked the court to make it official ... by legally recognizing him as the father. He also wants joint custody and for the court to determine how much child support he should pay.

Brittni then fired back with docs of her own, claiming it's not enough. She says Future left his kid without support for months. She's asking for that cash -- specifically from June and July -- plus future support.

Meanwhile Future's still battling his baby mama #1 (Jessica Smith) ... trying to shoot down her request to collect more cash for their 10-year-old son.

Yay fatherhood.