The correct option is
A late
In order to fill in the blank with the most suitable adjective or an adverb, let's understand what adjectives and adverbs are.Adjectives are the words that define/explain nouns or pronouns. For example: Roshni is pretty. Here, 'Roshni' is the noun and 'pretty' explains how she is, hence 'pretty' is the adjective.
Adverbs are the words that describe action words (verbs). For example: Ram fought bravely. Here, 'Ram' is the noun and 'fought' is the verb. 'Bravely' describes how he fought, hence it is the adverb.
NOTE: We need a word to describe the time and way she comes. Since 'comes' is a verb, the word that describes it is called an adverb. Hence, we need an adverb fill in the blank.
Option A - 'Lately' is an adverb, but it means recently. The sentence means that she is rarely on time. Thus option A is incorrect.
Option C - 'Lateness' is a noun and not an adverb. Hence, option C is incorrect.
Option D - 'Lateful' is a wrong word. Thus option D is incorrect.
Option B - 'Late' is an adjective and an adverb both. It can be used as an adverb here, which describes the time and the way she comes (reaches) for her lectures. Hence, option B is the correct answer.
'She often comes late for the lectures.'