Laser-plasma accelerators can produce a few GeV electron beams over a distance of a few cm. Such a beam typically has relatively low emittance, high peak current but a rather large energy spread and jitter. The large energy spread hinders the potential applications for coherent free-electron laser (FEL) radiation generation. We discuss a method to compensate the effects of beam energy spread by introducing a transverse variation of the undulator magnetic field. Such a transverse gradient undulator (TGU) together with a properly dispersed beam can greatly reduce the effects of electron energy spread and jitter on FEL performance. In this paper, we review the TGU concept, theory and discuss technical implementations. Using particle-in-cell simulations of a GeV laser-plasma accelerator and the FEL simulation code GENESIS that is modified to accommodate TGU, we show a soft x-ray FEL operating in the "water window" wavelengths can reach saturation with a undulator length of about 12 m. Show Partial Abstract