Freshly captured planets will drain your resources. You must upgrade your infrastructure on those planets, to get rid of that negative number. Otherwise you will crush your economy.

I was having difficulty with this myself... (I've only tried it so far with Vasari.)
I was playing with pirates off... (so far, I really don't like the pirate mechanic.)

I have not finished it, but I did get into positive territory.

What I did was:
- build only extractors, and capital ship factory.
- start upgrading the planet, spend all resources upgrading your planet until well into the green.
- build only a capital ship (that can colonize,) to clear and colonize the neighboring planets... but don't colonize them until you are well into the green on your first planet.
- upgrade your planet to 'home' planet.

I was able to really harrass the AI with my capital ship, cost him a fortune, and his second planet.

I quit the mission though, cuz I got on the phone, and didn't realize it wasn't paused, when I came back I had 11k credits, and lord knows how behind I was on the buildup.

In general I think this is a really badly done scenario. It seems there are very few successful strategic paths through the early game. You spend almost all your time waiting for the game, populations to build up, credits to build up... once you can financially expand, it's no longer relevant.

Yeah, only research the fleet capacity upgrade when you need it. I didn't realize at first that the upkeep was a function of my fleet capacity level, so I would upgrade that a whole bunch thinking it was smart to get it out of the way early.

while that will increase income, to reach its full potential u will need to put up a broadcast center :/

You had no homeworld so your planet starts with minimum Allegiance (25%). Once you get homeworld, your max Allegiance for the homeworld is 110%, but your current allegiance is still only 25, You need to raise it somehow, and the way to go about it is broadcast center. Note that allegiance multiplies your income. So at 25% you only get 25% income!

Yeah, that map is a pain in the butt.

The first thing you should do is purchase the Home Planet upgrade immediately. This brings you to the positive cash flow, by increasing your cash flow by 4 credits/second, and will increase allegiance over time. You should also get the first infrastructure upgrade as soon as you can. This brings your income up by another 1.5 credits/second. After that build your crystal extractor and the metal extractors. Build a scout or colonizer as needed to capture the two neutral extractors. This gives you an additional 1.2 metal/second. At this point your primary concern should be getting a capital ship out and about. From there you can capture your asteroid, which will auto-magically start at 90% allegiance, giving you an additional .4 metal and crystal per second. From there it is a normal map.

Yeah, when I first did that map as a newbie I played the entire thing without using the upgrade. The game lasted 3.5 hours, with me using the Vasari and their expensive ships. I was relying on the neutral extractors to provide metal I could sell on the market.

It was my first loss against the AI, when the Advent AI pushed on my two colonies with about 80 ships, including several capitals and a mass of Drone Hosts. My reward was seeing the artwork of the ship crashing down on the poor Vasari, as he raises his arms towards the heavens amidst the flame and death around him.