Dua asked in namaz e shab is fulfilled. My question is that when should we ask for our hajat in the prayer like after completion of it or in qunoot of namaz e witr or is there any specific dua after which we can seek for our needs?

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You may ask your needs in Qunoot but we should avoid in Qunoot to recite duas other than those mentiones in the Holy Quran or recommended by Aimmah (a.s).
But you can ask your Hajaat after completion of Salatul Layl. It won’t make any difference Allah (swt) know better but likes that His slave comes and ask from Him.