The temporary closure of the library will effect local patrons.

Affect- Verb, actions taken. Effect- Noun, to show the result

(A) will effect local patrons -> Incorrect use of "effect" (noun).
(B) will have an effect on local patrons -> "have an effect" is still using as noun. We need verb. Incorrect.
(C) will affect local patronage -> It is not showing result. Incorrect use of "affect" (verb).
(D) will affect local patrons -> "affect" is used as a verb. Let's keep it.
(E) has had an effect on local patrons -> has had is incorrect to use.

So, I think D.

The temporary closure of the library will effect local patrons.

(A) will effect local patrons
(B) will have an effect on local patrons
(C) will affect local patronage
(D) will affect local patrons
(E) has had an effect on local patrons

This question has a split over affect or effect. Effect is a noun and means results in whereas Affect is a verb.
Also patrons refers to the person who gives support while patronage is the amount of support given.

Now with this two in mind let us try to eliminate some choices.

(A) will effect local patrons
No verb in this choice as effect is not a verb
(B) will have an effect on local patrons
effect is an noun plus will have an effect is an indirect way of saying will affect.
(C) will affect local patronage
Patronage is incorrect here as this means that it will affect the amount of money given. Well it might affect future donations but can not affect the already given amount
(D) will affect local patrons
Can not see anything wrong here
(E) has had an effect on local patrons
has had an affect. We don't need a present perfect here.We are trying to predict something that might happen because of this closure
_________________

"effect" here has different meaning
Eliminate A,B &E

usage of "patronage" is wrong in C

Ans D