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The fused relative word 'what' generally means "the thing(s) that/which".

But there are some instances where this 'what' seems to refer to "person(s)" as in these examples:

(1) From a Bustle article titled "Julián Castro On Exploring A 2020 Run & The Possibility Of Becoming Our First Latino President":

Bustle's Alicia Menendez spoke with Castro about his decision to explore running, the stakes of the election, and the possibility of becoming America’s first Latino president.

Alicia Menendez: You are the first of what many anticipate will be about a dozen Democratic hopefuls to officially announce an exploratory committee. Why do it now?

(2) From a CNBC article titled "America’s foreign policy is seen threatened by James Mattis’ exit, feeling of chaos in Washington":

Mattis was seen as the lone remaining grownup in Trump’s Cabinet, willing to push back against a commander-in-chief who disdains the government’s foreign policy apparatus, and has little use for traditional diplomacy.

“Secretary Mattis represents the last of what we might call the mainstream foreign policy thinkers in the Trump administration,” said Jim Lindsay, who recently co-authored the book The Empty Throne: America’s Abdication of Global Leadership. “His departure is going to shape the balance of advice the president gets. And I think it is also going to change how American foreign policy is viewed overseas.”

In the above examples, does the 'what' mean "the person(s) that/who" instead of "the thing(s) that/which"?

Then word “what” refers to an assumed outside perception of the issue being discussed.

Using Julian Castro example, Alicia Menendez: You are the first of what many anticipate will be about a dozen Democratic hopefuls to officially announce an exploratory committee. Why do it now?

Omit the “what” part of the sentence to see the assumed perception.

This is the assumed outside perception: “You are the first Democratic hopeful to officially announce an exploratory committee.”

To answer your question, it would be the person. Julian Castro is the assumed Democratic hopeful.

Let’s look at Mattis.

Omit the what part of the sentence.

Secretary Mattis represents the mainstream foreign policy thinkers in the Trump administration.

Its all about Mattis. He’s the assumed perceived policy thinker. So it would the person.

Those are long sentences. Let's simplify things so that the example isn't so long as the ones you've given:

The Cretaceous is the last period of what we call the Cenozoic era.

That was the initial problem of what became an ordeal.

Is "what" referring to what comes before or after it?

I would claim that the pronoun "what" refers to the entire phrase beginning from "what", including itself (why it can be called fused) onward. To quote the relative pronoun article of Wikipedia:

For example, in "I like what you did", what is a relative pronoun, but
without an antecedent. The clause what you did itself plays the role
of a nominal (the object of like) in the main clause. A relative
pronoun used this way is sometimes called a fused relative pronoun,
since the antecedent appears fused into the pronoun (what in this
example can be regarded as a fusion of that which).
Relative
pronoun, antecedents

Note the specifically that first it says it has no antecedent, and then says that the antecedent "appears fused into the pronoun". In its explanation of "free relative clause" in the "Relative clause" article it says the following:

A free relative clause, on the other hand, does not have an explicit
antecedent external to itself. Instead, the relative clause itself
takes the place of an argument in the matrix clause. For example, in
the English sentence "I like what I see", the clause what I see is a
free relative clause, because it has no antecedent, but itself serves
as the object of the verb like in the main clause. (An alternative
analysis is that the free relative clause has zero as its antecedent.)
Relative clause, bound and free

So from Wikipedia we have the claims that in such constructions the relative pronoun "what" either:

Has no antecedent.

Has zero as its antecedent.

The "antecedent appears fused into the pronoun".

So in both of your examples the "what" pronoun refers to everything from itself onward (in the case of the second example where the quotation ends).

Can the relative word 'what' mean “the person(s) that”?

In the same construction as the ones given above:

Leonardo da Vinci was a prime example of what we call a polymath.

A polymath is a person, I'd answer your question with a yes.

Also, there is some objection to using "that" instead of "who" as a relative pronoun in sentences like:

All the parents that were present waited to see the teachers.

Although there are recommendations that strictly say to use "who" for people and "that" for things, there is a variation of opinion. And to the extent that the relative pronoun "what" means "that which", as is listed in many dictionaries, then there may be an objection on this ground.