Holographic speckle is a major impediment to computer-generated holographic (CGH) projections in applications ranging from display, optical tweezers, and machining to optogenetic neural control. We present an iterative phase retrieval algorithm that allows the projection of amplitude-controlled speckle-free one-dimensional patterns with a high degree of pattern uniformity. The algorithm, termed the weighted Gerchberg–Saxton with phase-control (GSW-PC), is shown to have the ability to simultaneously control both the phase and amplitude of projected patterns with high diffraction efficiencies. Furthermore, we show that the framework can address the challenge of projecting volumetric phase and amplitude-controlled patterns, by incorporating GSW-PC with the angular spectrum method. The algorithms’ performance is numerically and experimentally tested, and further compared with conventional and modern CGH techniques.