The correct option is C Pull through
A phrasal verb is an idiomatic phrase made up of a verb and another element, usually an adverb or a preposition (verb + adv., verb + prep.).
The given sentence means that the person 'she' is ill, or she is in despair.
Option A: 'Come out' means 'to stop being fixed somewhere or in some situation'. This phrase must be followed by a few more words in order to complete the meaning, or else it does not complete the given sentence.
(For example: I hope she will come out of her misery.)
Option B: 'Pull out' as a phrasal verb means 'to leave'. It isn't relevant to the statement, hence B is incorrect.
Option D: 'Go out' as a phrasal verb means 'to go on a date with someone', or 'to leave the house and go somewhere'. Both meanings aren't relevant here, hence D is incorrect.
Option C: 'Pull through' means 'to manage or succeed in a difficult situation'. It also means 'to stay alive'. Here the words "she has been lying in bed for the last fortnight" refer to some unknown situation. Hence this is the most suitable phrase and option C is correct.