The acclaimed writer said that his writing, which were power

it has to be B : his writing was influenced by powerful storytellers in his family
plz confirm the source and the OA

this gives a wrong sense :powerfully influenced

B is a trap. If you pay attention to the question's structure, you will see the use of passive voice is incorrect.

B says:
The acclaimed writer said that his writing his writing was influenced by powerful storytellers in his family that the landscape of his childhood also shaped his thinking and provided themes for his stories.

The structure is: X said that .... passive voice...... and that... active voice.
The structure is not parallel. C is much better.

Hope it helps.

Who can explain why is powerfully influenced OK and I should have not gone after something like powerful influence?
Would option D be preffered if it said 'storytellers in the family had powerful influence on his writing'?

Isn't idiomatic usage is 'influence on X'?

The acclaimed writer said that his writing, which were powerfully influenced by storytellers in its family and that the landscape of his childhood also shaped his thinking and provided themes for his stories.

(A) his writing, which were powerfully influenced by storytellers in its family
writing is singular, and hence cannot be followed by were. Another error is with 'its', which has no referent.

(B) his writing was influenced by powerful storytellers in his family
A keen observation reveals that there are two relative pronoun clauses introduced by the relative pronoun - that. The second relative pronoun clause is in active voice, and the first one is in passive. Grammatically correct though. (Don't eliminate, let's look for better options)

(C) storytellers in his family powerfully influenced his writing
The issue in B is corrected here. The relative pronoun clause is in active voice. The problem arises with 'powerfully'. 'Powerfully' is an adverb modifying the verb 'influenced'. Looks fine. Even if you feel that usage of powerfully is wrong here, substitute, powerfully with strongly - strongly influenced his writing - looks good. This option is clearly better than B.

(D) storytellers in the family being powerful influence on his writing
Too many logical issues here. 'the family' doesn't state 'whose' family. 'being powerful influence' is wrong usage. It could have been 'been a powerful influence on his writing'. 'Being' is the present continuous form of 'be' and the usage of the present continuous form in the context is wrong. Eliminate.

Note: Always verb form is preferred over an equivalent noun form.

His family powerfully/strongly influenced his writing.
His family had a powerful/strong influence on his writing

(E) powerfully influential in his writing were family storytellers
Passive voice. Meaning is distorted. Eliminate.

A. Writing Was, not Were. A is out.
D. There has to be an independent clause after "that" in underlined part, but we have not it here. D is out.
E. Strange construction. Reversed order of the words. Adjective "influential" instead of the verb. Lack of parallelism in two parts after two "that". E is out.
B. GMAT likes such mini traps. It sounds good. But we have passive voice after first "that" and active voice after second "that". It makes B not parallel. B is out.

C is good.

ERROR ANALYSIS -

1) S-V doesn't agree in number
2) Absence of verb in underlined portion

ANSWER CHOICE ANALYSIS -

A) S-V doesn't agree in number, absence of verb in underlined portion
B) powerful storytellers - is wrong, story was 'powerfully influenced'
C) CORRECT
D) Use of 'being' is correct only if 1) it is used as a noun 2) used in passive continuous tense
E) Passive voice used