ministry vs Home Office

ministry and Home Office both are nouns.

Word	Noun	Adjective	Verb	Adverb
ministry	Yes	No	No	No
Home Office	Yes	No	No	No

As nouns, Home Office is a hyponym of ministry; that is, Home Office is a word with a more specific, narrower meaning than ministry:

ministry: a government department under the direction of a minister of state
Home Office: the government department in charge of domestic affairs

Other hyponyms of ministry include Foreign Office.

ministry (noun)	Home Office (noun)
the work of a minister of religion	the government department in charge of domestic affairs
building where the business of a government ministry is transacted
a government department under the direction of a minister of state
religious ministers collectively (especially Presbyterian)