The most important thing to do is... LEAVE THE BODIES WHERE THEY LIE.
Had a problem with pigeons. Bought a .177 springer. Killed about a dozen in about a week's time. As I live next to about 20 acres of vacant land, I just tossed the bodies into the brush as coyote food and didn't give it another thought. After that first week, pigeons became a rare sighting within range of my house. They were still all over the neighborhood, but not near MY house.
The next spring, they returned. I killed about 3-4 in a weekish and they disappeared again...despite the fact that my next door neighbor throws out seed for birds every morning and evening. Other birds? You bet. Pigeons? Not a one.
That was probably 4-5 years ago. There are still pigeons all over the neighborhood... But there still are none within pellet gun range of my house. I don't pretend to know how they teach their young, but somehow they do.
What's this got to do with leaving bodies?
OK, the second half of the story... I have a coworker who had a similar problem. He killed pigeons constantly with no relief. I found it odd that the pigeons in my neighborhood got smart while his did not. I commented on it to somebody else I knew. Their response: "What does he do with the bodies? I bet he cleans them up and throws them away... Dead pigeons are a wonderful scarecrow for living pigeons."
I asked coworker what he did with the bodies. "I throw them in the trash," was his reply. I relayed the other individual's comments and told him to leave the bodies where they fell.
Within 2 weeks coworker reported that pigeons were no longer assulting his house.