The acclaimed writer said that his writing, which were power

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.