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Prolonged unemployment can create serious psychological problems that, in the long run, actually contribute to continued jobless. In a society that stresses the relationship between productive work and personal value, it is easy enough to equate long-term unemployment with personal worthlessness. That is, in fact, precisely what many unemployed men and women begin to do. Out of a job for a year or more, they begin to see themselves as worthless human beings without any value to society. In what amounts to a vicious cycle, their sense of personal worthlessness further diminishes their chances of gaining employment. Sometimes they stop looking for work altogether, sure in their despair that no one will hire them.

I'd like to know if it is right that "in" means "during" and "what" means "the period."
And I'd like to know if "sure" modifies the whole "in" prepositional phrase immediately following it.
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Comments  
"what amounts to a vicious cycle" means "a thing/situation/circumstance that amounts to a vicious cycle". "in" could be read as meaning that those people are in the situation described. Alternatively, and less literally, it could be read as presenting "what amounts to a vicious circle" as a description of the state arising from "their sense of personal worthlessness further diminishes their chances of gaining employment". I most naturally understand it as the latter.

The last phrase can also be read in two ways. It could mean "sure that no one will hire them", with "in their despair" describing their state at the time. Alternatively, "sure in" could indicate that they see "their despair that no one will hire them" as a fixed and inflexible state.
park sang joonIn what amounts to a vicious cycle,
In what is basically a vicious cycle, their sense ... etc.
or
As they are basically in a vicious cycle, their sense ... etc.
park sang joon sure in their despair that no one will hire them.
... as they feel sure (in that state of desperation) that no one will hire them.
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park sang joonsure in their despair that no one will hire them.
sure that no one will hire them (because they are in despair).

'that no one will hire them' is the complement to 'sure'. 'in their despair' is more like an adjunct explaining why they are sure, so it modifies the whole 'sure that ...' AdjP, which in turn modifies the preceding clause, giving a reason why they stop looking for work.
park sang joonIn what amounts to a vicious cycle
'what' is a fused relative; it contains both antecedent and relative word ('that' and 'which') in one word 'what' = 'that which'.

In a vicious cycle > In that (thing) which amounts to a vicious cycle > In what amounts to a vicious cycle

Your paraphrase 'during the period amounts to a vicious cycle' will not work at all. 'in' should not be taken completely literally here.
___________

This structure is worth becoming familiar with. It's a way of avoiding the claim that a thing actually is exactly that thing. It's a way of saying that maybe the word choice isn't exactly correct.

a mystery > what amounts to a mystery; what is called a mystery; what we might call a mystery; what many people may believe is a mystery

These expressions can crowd out the rest of the sentence and therefore seem much more important than they really are in the structure of the whole sentence.

Today I received what I laughingly call my weekly paycheck. In effect, the structure is just I received my paycheck.

More examples found online:

One powerful answer, we found, is in [what we like to call] innovation communities.
He is a throwback in [what we will call for the millionth time] the Age of Twitter.
I believe that there's a lot of protectionism in [what we call] the First World.
The delegates have become bit players in [what amounts to] a ritual drama.
The language is code for [what amounts to] a German veto on further bond purchases.
Still, it's refreshing to see brands take a risk for [what seems like] a good cause.
We are getting fed up with being blamed for [what is primarily] a US made problem.

CJ
In what amounts to a vicious cycle.

Is "what amounts to a vicious cycle" an object of the preposition?
Another example: "The issue now becomes whether Tsipras was right to decide not to invoke this plan in the face of what amounted to extortion from the creditors." (Paul Krugman.)
Students: Are you brave enough to let our tutors analyse your pronunciation?
tkacka15In what amounts to a vicious cycle.Is "what amounts to a vicious cycle" an object of the preposition?
Yes.
tkacka15Another example: "The issue now becomes whether Tsipras was right to decide not to invoke this plan in the face of what amounted to extortion from the creditors." (Paul Krugman.)
Good example. Emotion: smile

CJ
Thank you, GPY, teechr, and Mr Jim, for your very helpful answer. Emotion: smile

Of my example:
1. Sometimes they stop looking for work altogether, sure <in their despair> that no one will hire them.

2. Sometimes they stop looking for work altogether, (tery are) sure in their despair that no one will hire them.
3. Sometimes they stop looking for work altogether, (Being) sure in their despair that no one will hire them.
Then I'd like to know if I can regard #1 as either #2 or #3.
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