Which word should fill the gap: "In ____________, I would like to add my congratulations to your daughter and her husband on the birth of their second child."?

The correct choice is "addition" because it serves as a noun in this context. The phrase "In addition" is commonly used to introduce an extra piece of information or to express a supplementary idea. It effectively connects the statement about congratulating the daughter and her husband with the broader context of the conversation.

The other options do not fit the grammatical structure required in this sentence. "Add" is a verb and does not provide the necessary context for supplementing information. "Additive" is an adjective that describes something that can enhance or increase another substance but doesn't fit as a transition phrase. "Adding," the gerund form, also does not function properly in this sentence, as the phrase requires a noun to act as a coherent part of the introductory clause. Thus, "addition" is the only word that appropriately completes the sentence.