The correct option is C While going for a job
An adverb clause is a group of words containing a subject and a verb,
and it functions as an adverb, by modifying an adjective, verb or an
adverb. Adverb clauses generally begin with a subordinate conjunction
(since, because, while, etc.)
Generally, while reducing an adverb clause to a phrase, the subject is removed and the verb is changed to a present participle (verb + ing). (If there is an auxiliary/helping verb attached, it is removed.)
The underlined words form a clause,
with the adverb of time "while", the subject "she", verb "went" and
object "for a job". Hence the sentence needs to be changed and option A is
incorrect.
Option C is correct because the subject is removed, hence
modifying the adverb clause into an adverb phrase (with the adverb of
time intact).
Option B is incorrect because the adverb of time is removed, hence not making it an adverb phrase.
Option D is incorrect because it's only a word, not a clause/phrase.