In South Africa the baboons not only annoy farmers by raiding cornfields but also chicken coops.

Question

Rewritten:
In South Africa the baboons not only annoy farmers by raiding cornfields but also annoy chicken coops. Hmmm! "In South Africa, the baboons annoy farmers by raiding cornfields and the chicken coops."

do you put "the chicken coops" or just "chicken coops"???

Just my $.02 here.

In South Africa the baboons not only annoy farmers by raiding cornfields, but also by raiding chicken coops.

I'm doubtful the article 'the' is needed, unless there are specific chicken coops you have in mind.
...but take Writeteacher's word as an authority here. Hope this helps.