All 5 Uses

despair
in
The Railway Children
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He had often acted as veterinary surgeon to the wooden rocking-horse; once he had saved its life when all human aid was despaired of, and the poor creature was given up for lost, and even the carpenter said he didn't see his way to do anything.†
p. 5.4

despaired = lost hope
'Only, please, you mustn't walk on the line.'
'Not if we face the way the train's coming?' asked Peter, after a gloomy pause, in which glances of despair were exchanged.
p. 53.7

despair = hopelessness
"Oh, don't!" said Phyllis, in despair; "this is truly dreadful!"
p. 106.7 *

despair = distress
Perks sat down heavily in the elbow-chair and looked at them with what Bobbie afterwards described as withering glances of gloomy despair.
p. 173.6

despair = hopelessness
"He's fallen down in a fit, more likely," said Phyllis, "all human aid despaired of."
p. 250.6 *

despaired = lost hope

Definitions:

(1) (despair as in: she felt despair)
hopelessness; or distress (such as extreme worry or sadness from feeling powerless to change a bad situation)
(2) (despair as in: do not despair)
lose hope or feel distress
(3) (despair as in: she was the despair of the team)
something that causes hopelessness or great distress
(4) (meaning too rare to warrant focus)