They worked with all sincerity. The underlined phrase is-

An adverbial phrase is simply a group of two or more words that functions as an adverb in a sentence.
Just as an adverb can modify a verb, adjective or another adverb, an adverbial phrase can further describe a verb, adverb or adjective.
এখানে with all sincerity (সম্পূর্ণ আন্তরিকতা সহকারে) phrase টি ‘worked’ verb টিকে modify করছে।
তাই এটি Adverbial phrase বলে গণ্য হবে।