Comparing (C) and (E), we essentially have:

(C) The overall slackening of growth in productivity is influenced... by the coming to an end of a period.
(E) The overall slackening of growth in productivity is influenced... by the end of a period.

When we use "end of [something]", we are usually referring to a specific time or place. For example,

We reached the end of the trail.
Please submit your expense report by the end of the workday.
The end of the Permian Period coincides with the end of the Paleozoic era.

On the other hand, the "coming to an end of [something]" refers, not to the end itself, but to the approach of the end.

Choice (E) refers to the actual end of the period. Choice (C) refers to the approach of the of the period. So which makes more sense?

For one thing, the verb "is influenced" suggests that the "influencing" is an ongoing action that is happening right now. The end of a period is a specific point in time. It doesn't make as much sense to say that the end itself IS influencing the slackening.
On the other hand, the "coming to an end of a period" is an ongoing action. It makes perfect sense to say that the approach of the end IS influencing the slackening.

(C) is a bit better meaning-wise as well. It's not the end itself but the approach of the end that influences the slackening.

Also, to say that growth is "rapid" is a fairly vague and subjective description. Who's to say when exactly the "rapid" growth ends? When do we go from "rapid" growth to "kinda fast" growth to "slow growth" to "no growth", etc.? If we go with choice (E), we are referring to the specific end of something that is, by nature, not so easily defined.

Choice (C) is more flexible. When exactly will that period end? Has it already ended?? It doesn't matter. The growth is slowing, and the period of rapid growth is coming to an end.

Because (C) is clearer and more logical, it's our winner.