Do I gain income for connecting city-states to my trade network?

City-States keep asking me to connect to them, but I'm not sure if the life-long upkeep is worth the one-time influence gain. If, on the other hand, I get influence and an income boost, I'm gonna have to buy my workers an extra set of shovels and connect to city-states like mad.

I could, of course, just erase the roads after I make them, but that seems like a step away from the narrative of the game and the reason behind making the road in the first place.

Thoughts?

No, city states do not give you trade income.

Now I've logged a couple hundred hours including the latest patch (which change a lot) and IMO:

1. The costs of maintaining a road to your loving city-state far outweighs the benefit in the long run.
2. I have removed roads as suggested earlier. Meh.
3. I had a match where it was going to take 5 tiles to connect to a city state onto the direct road, but only two from the city on the other side.. it was connected solely by a harbor for its trade route. The road I built, and Helsinki built their end of it.. did NOT trigger the expected bonus,