English Problem

I'm not sure if the following is correct terms to describe the types of modifier, but I guess the following may give you some direction:

Q1. "Hong Kong's" is the modifer for "problem", I think it is called a possessive form. Meaning using XXX's to describe the problem "belongs" to Hong Kong.

Q2. "mass" is the modifer for "media", I think it may be called a collective form or quantitative modifier to describe the media is not a specific type but the whole "mass".

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(In grammar,
a modifier (or qualifier) is an optional element in phrase
structure or clause structure;
the removal of the modifier typically doesn't affect the grammaticality
of the construction. Modifiers can be a word, a phrase or an entire
clause. Semantically, modifiers describe and provide more accurate
definitional meaning for another element.)

Sentence 1: He said Hong Kong's problem is how to enjoy freedom of
speech while not abusing it.
Question 1: which type of modifier is
used with the noun "problem" in sentence 1?
The modifier "Hong Kong's" is used with "problem" as an adjective in noun phrase type. (Hong Kong's is a proper adjective.)

Sentence 2: we can also measure how healthy the mass media is.
Question
2: What is the modifier for the word "media" in sentence 2?
The modifier "mass" is used with "media" as a noun in noun phrase type.

There is no modifier in both Q.1 and Q.2 for the words you inquire about.
Do you mean
1. Which type of modifier can be used with the noun "problem" in sentence 1?
2. What is the modifier to use for the word "media" in sentence 2?