The correct option is B While I had my green tea
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 expanding an adverb phrase to a clause, the subject is added and the present participle (verb + ing) is changed into a verb. (If there is an auxiliary/helping verb required, it is added.)
The underlined words form a phrase, it has no subject. Hence the sentence needs to be changed and option A is
incorrect.
Option B is correct because the subject "I" is added and the verb "had" replaces the present participle.
Option C is incorrect because it is a coordinate clause, whereas an adverb clause is subordinate.
Option D is incorrect as "because" is an adverb of reason, whereas the adverb here pertains to time.