The word "hope" can be used to refer to a couple of (similar, but slightly different) things:
"A hope", or "hopes", as a countable noun generally refers to one's wishes, dreams or specific aspirations for the future. Since these "hopes" are specific desires, this form is often used with attached descriptive clauses such as "hopes of <doing something>" or "hopes that <something> will happen", such as in your examples:
She has hopes of studying to be a nurse (LDOCE)
he secretly cherished hopes that George would marry his daughter (OALD)
"Hope" as an uncountable noun refers to a more general feeling of optimism or positive outlook for the future (and is used similarly to other emotions, like "joy" or "sadness"). This is more commonly used by itself without supporting phrases, such as in your examples:
Hope faded after wrecked remains of the ship were washed onto the shore.
The treatment gave him renewed hope.
Now, as you may notice, there are some cases where "hope" is used with qualifiers, and some cases where "hopes" is used by itself, etc, so these aren't unbreakable rules, just the usual practice.
Additionally, there are some set phrases that may use one or the other even though it seems like the other one might be more fitting, for example:
Hold out hope (of something occurring)
Give up hope (that something will occur)
These generally refer to one's general optimistic outlook (emotion) but specifically about a particular thing happening, so it's sorta a cross between the two above cases. In these set phrases, the uncountable "hope" is used even though it may be about a specific desire (so, for example, saying somebody should "give up hopes" would sound rather strange, and while "give up hopes of <something>" might actually technically be OK, it would be more common for somebody to say either "give up hope of <something>" (using the set phrase) or "give up <his/her/your/etc> hopes of <something>" (avoiding the similarity with the set phrase entirely) instead).
High hopes / hopes are high
This phrase is generally used to indicate somebody has a lot of "hope" (the general, uncountable kind), but the set phrase uses the word "hopes" anyway (and saying somebody has "high hope", would sound strange to most people).