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This paper considers short-term power constrained cell-free massive multiple-input multiple-output (MIMO) scenarios where a large set of multi-antenna access points (APs) provide service to a group of single-antenna mobile stations (MSs) on a spatially correlated multipath environment. Based on a probabilistic approach, the spatially correlated propagation links are modeled using either Ricean or Rayleigh fading channel models that combine a deterministic line-of-sight (LOS) propagation path with a small-scale fading caused by non-line-of-sight (NLOS) multipath propagation. Assuming the use of minimum mean square error (MMSE) channel estimates, closed-form expressions for the downlink (DL) achievable spectral efficiency of a cellfree massive MIMO network with short-term power constraints (i.e., a vector normalized conjugate beamformer (NCB)) are derived and benchmarked against that provided by the conventional cell-free massive MIMO network with long-term power constraints (i.e., the conventional conjugate beamforming (CB)). These expressions, encompassing the effects of spatial antenna correlation, Ricean/Rayleigh fading and pilot contamination, are then used to derive both pragmatic and optimal max-min peruser power allocation strategies and to gain theoretical insight on the performance advantage provided by the use of short-term power constraints instead of the conventional long-term power constrained approach.