Мой муж сейчас в офисе и пишет отчёт.

My husband is now in the office and is writing a report.

Breakdown of Мой муж сейчас в офисе и пишет отчёт.

• мой: my
• писать: to write
• в: in
• и: and

Questions & Answers about Мой муж сейчас в офисе и пишет отчёт.

Why is there no word for “is” in Мой муж сейчас в офисе?

Russian usually omits the verb “to be” (быть) in the present tense when it simply links a subject and a place/description.
So Мой муж сейчас в офисе literally is “My husband now in (the) office”, but it means “My husband is in the office now.”
You only see forms of быть in the past (был, была, было, были) and future (будет, будут, etc.), or in special emphatic/grammatical constructions.

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Why is it мой муж and not something like меня муж?

Мой is the possessive pronoun “my” in the nominative masculine singular, matching муж (husband), which is also nominative masculine singular.
Меня is the genitive/accusative form of я (I/me) and cannot be used as a possessive here.
So мой муж = my husband (subject of the sentence).

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Could it be моего мужа instead of мой муж?

Моего мужа is the genitive or accusative form and is used when “my husband” is not the subject, but an object or after certain prepositions.
Here, муж is the subject (the one doing the action), so it must be in the nominative: мой муж.
Example contrast: Я жду моего мужа – I am waiting for my husband (object, accusative).