Some data strongly supports the idea that if a primary pair of coyotes are killed, the group breaks up and reproduction actually increases.
And when you look at the best GPS-collar data available, you realized what a no-win situation kill/trapping coyotes really is. Only about 50% of the coyotes you see in a given area are "resident" coyotes. The other half are drifters. Kill the drifters and you've done nothing to reduce the local population. Kill the residents and the drifters take over and become residents.
About the only data I've seen that suggests you can have an impact on the coyotes' predation is to hit them hard just before the birth cycle of the prey animal you are concerned with. In essence, hit coyotes hard in late spring and you can reduce their impact on fawn and turkey poult predation temporarily (that year). But the coyote population quickly bounces back. So you would have to hit the 'yotes hard EVERY spring.