In this paper, we generalize the conditions for the exact or stable recovery of weighted k-sparse signals in weighted sparse phase retrieval in our previous work [11] from the weighted \(\ell _1\) minimization to the weighted \(\ell _q\, (0<q\le 1)\) minimization in a broad sense. Specifically, we first present that the weighted null space property (WNSP) is a sufficient and necessary condition to guarantee the exact recovery of a weighted k-sparse signal from its noiseless phaseless measurements via the weighted \(\ell _q\, (0<q\le 1)\) minimization in both the real and complex cases. In addition, we establish a general strong weighted restricted isometry property (SWRIP) condition for the stable recovery of a weighted k-sparse signal from its noisy phaseless measurements via the weighted \(\ell _q\, (0<q\le 1)\) minimization in the real case.
