In mathematical statistics, the following problem (simple hypothesis test) is considered: given a data sample, test the hypothesis $H_0$ stating that all sampled values are values of a random variable with a given distribution $P_0$ against the hypothesis $H_1$ stating that all sampled values are values of a random variable with another given distribution $P_1$.

The question I am dealing with is a bit more general: test the same hypothesis $H_0$ against the hypothesis $H_1'$ stating that some of the sampled values are values of a random variable with a given distribution $P_0$, and some others are values of a random variable with another given distribution $P_1$.

The original question is usually settled using Neyman-Pearson Lemma (https://en.wikipedia.org/wiki/Neyman-Pearson_lemma). What are the methods available for the more general hypothesis test? I know mathematical statistics from a postgraduate qualifying exam, but I am not a specialist. I need this for applied statistical problems I am solving, and I failed to find an answer in available textbooks and monographs. Can you direct me to some approaches?