This paper explores covariance estimation from energy measurements that are collected via a quadratic form of measurement vectors. A popular structural model is considered where the covariance matrices possess low-rank and sparse structures simultaneously. We investigate a weighted convex relaxation algorithm tailored for this joint structure, which guarantees exact and universal recovery from a small number of measurements. The algorithm is also robust against noise and imperfect structural assumptions. In particular, when the non-zero entries of the covariance matrix exhibit power-law decay, our algorithm admits exact recovery as soon as the number of measurements exceeds the theoretic limit. Our method is related to sparse phase retrieval: the analysis framework herein recovers and strengthens the best-known performance guarantees by extending them to approximately sparse and noisy scenarios as well as a broader class of measurement vectors, and our results are derived using much simpler analysis methods.