At our last house we had a small bathtub-sized pond filled by rain-barrel water. We didn't get mosquitos in it, and it actually ended up attracting insects like dragonflies that eat mosquitos, so they would fly around and eat mosquitos all over our property. We achieved this by doing two things:
1. My wife bought these things called "mosquito dunks" -- it's a brick of some kind of material that you toss into the pond. The brick disintegrates and kills all mosquito larvae, yet somehow leaves other insect larvae intact and also leaves the water safe for other wildlife to drink/live in/etc. We only had to add them a few times near the beginning when we were establishing the pond. (Do a DuckDuckGo search on that term: "mosquito dunks".) After that, part 2 took over:
2. We designed the pond so there were flat rocks and moss all around the edge of it, and I let the grass all around it in about a 6-12" perimeter grow super tall. This created an ideal habitat for pond spiders, dragonflies, and other creatures that eat mosquitos. (This made the pond super fun to just sit and watch; there was always a lot of activity.) We also took a bunch of old bark, made it into a pile near the pond so that each piece of bark curled downward leaving a hollow underneath, then covered the pile in some of the sod we'd removed to dig the pond. That was supposed to be a habitat to attract frogs or toads, who are also renowned mosquito-eaters, but none ever came. Allegedly "if you build it they will come" but evidently, like with a Tesla, "your mileage may vary."
Oh, one more tip for rain barrels, if you have the stomach for it. This was something a former coworker of mine used to do: keep goldfish in your rain barrel instead of screening over top. The fish will eat any mosquito larvae (as well as algae maybe?) and poop into the bottom of the barrel. The fish-poop mixture at the bottom of the barrel creates water that is like steroids for plants. It's really good for them, I mean. You do have to get the fish out of there and keep them inside for winter though, if you live somewhere cold.
Now, I lost touch with this coworker and forgot to ask her what kind of roof she had on her house. I wonder if the tiny asphalt or fiberglass kernels that wash off the roof and into the rain barrel would harm the fish. Maybe she had a metal or stone/ceramic roof so that wasn't an issue. (Probably not based on where she lived, but who knows.)