I recommend not "investing" in a city-state unless you can get some influence for free to start out. Killing Barbarians or doing another quest for them will get your foot in the door, and let the $250 you have to spend periodically give you a lot more value. For instance, that $250 will be keeping you above the "Allied" line instead of just barely making you friends with the city-state.
Likewise, as mentioned above, the Patronage tree will give you significant bonuses to your influence over city-states.
It seems that the design is such that in the early game you won't be able to guarantee a city-state ally without committing a lot of cash.
To elaborate (not 100% sure on all these numbers but the overall point is still valid):
If you spend $250 to become Friends with a city-state, starting from 0 influence, you'll get 35 influence, making you barely Friends (30 influence). This means you'll be friends for 5 turns, at which point you'll need to spend another $250 to be friends again. You probably don't have another $250 that soon, though, so the influence runs all the way down close to zero before you're able to spend another $250, which again only gives you a few turns of friendship. This is obviously unsustainable.
If instead you perform a mission for the city-state and get 30 or 40 influence to start with, every time you spend $250 you'll get to use all 35 points of that influence before needing to spend another $250.
Similarly, you'd be better off saving your money to start with $500 worth of influence than buying $250 and not being able to afford more right away.
Long story short: if you're already friends, you get more value for your money. Likewise, if you're already allies, you get even MORE value for money you spend to sustain at that level.
I've been pretty happy with automated workers so far. One thing they will do is adjust things dynamically as your cities grow. They'll swap trading posts for farms if the city needs more food and that kind of thing.
If your cities are efficient you probably will end up with trading posts on nearly all non-hill tiles, though, which is a little boring-looking.
One further note: I did notice that the auto-workers built a trading post on a marsh without actually clearing the marsh (which could ONLY help the city), so that wasn't ideal.
Best Answer
There are at least three ways this can happen:
It looks like Hanoi at least you are at war with.