The correct option is B up
Phrasal verbs are phrases, formed with a verb and a preposition or an adverb. The meaning thus formed is different from its literal meaning.
Option B: The preposition 'up' forms a phrasal verb with 'put' and 'with'.
'Put up with' is a phrasal verb that means 'to accept an unpleasant situation or experience (or person).
The meaning is relevant to the given sentence, as it means: Ram Nagar will have to accept that there will be water shortage for a few more weeks (which is an unpleasant situation). Hence option B is incorrect.
Option A: 'Put in' means 'to give effort to something'.
Option C: 'Put on' means 'to get dressed'.
Option D: 'Put out' means 'stop something from burning', like 'put out a fire'.
'Put in', 'put on', 'put out' are all phrasal verbs, but they do not form phrasal verbs with 'with'.
As a result, the meaning of 'put in with', 'put on with' and 'put out with' are literal as well as vague. They do not form a coherent meaning in the sentence.
Hence options A, C and D are incorrect.