In the few-cycle pulse regime of laser-cluster interaction (intensity > 10(16) W/cm(2), wavelength > 780 nm), laser absorption is mostly collisionless and may happen via anharmonic resonance (AHR) process in the overdense (cluster) plasma potential. Many experiments, theory and simulation show average absorbed energy per cluster-electron (epsilon(A)) close to the electron's ponderomotive energy (U-p) in the collisionless regime. In this work, by simple rigid sphere model (RSM) and detailed particle-in-cell (PIC) simulation, we show enhanced epsilon(A) approximate to 30-70U(p)-a 15-30 fold increase-with an external (crossed) magnetic field near the electron-cyclotron resonance (ECR). Due to relativistic mass increase, electrons quickly deviate from the standard (non-relativistic) ECR, but time-dependent relativistic-ECR (RECR) happens which also contributes to enhanced epsilon(A). Here laser is coupled to electrons in two stages, i.e, AHR and ECR/RECR. To probe further we retrieve the phase-difference Delta psi between the driving electric field and corresponding velocity component for each electron (in PIC and RSM). We find absorption by electron via AHR happens in a very short interval Delta tau for less than half a laser period where Delta psi remains close to pi (necessary condition for maximum laser absorption) and then Delta psi drops to its initial pi/2 (meaning no absorption) after such short-lived AHR. On the contrary, auxiliary magnetic field near the ECR modifies AHR scenario inside the cluster and also helps maintaining the required phase Delta psi approximate to pi it for the liberated cluster-electron accompanied by frequency matching for ECR/RECR for a prolonged Delta tau (which covers 50-60% of the laser pulse through pulse maxima) even after AHR-leading to jump in epsilon(A) approximate to N 30-70U(p). We note that to realize the second stage of enhanced energy coupling via ECR/RECR, the first stage via AHR is necessary.
