Let $ u \in L_{sp} \cap C^{1, 1}_{\rm loc}(\mathbb{R}^n\setminus\{0\}) $ be a positive solution, which may blow up at zero, of the equation
$ (-\Delta)^s_p u = \left(\frac{1}{|x|^{n-\beta }} * \frac{u^q}{|x|^\alpha}\right) \frac{u^{q-1 }}{|x|^\alpha} \quad\text{ in } \mathbb{R}^n \setminus \{0\}, $
where $ 0 < s < 1 $, $ 0 < \beta < n $, $ p>2 $, $ q\ge 1 $ and $ \alpha>0 $. We prove that if $ u $ satisfies some suitable asymptotic properties, then $ u $ must be radially symmetric and monotone decreasing about the origin. In stead of using equivalent fractional systems, we exploit a direct method of moving planes for the weighted Choquard nonlinearity. This method allows us to cover the full range $ 0 < \beta < n $ in our results.
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