A high repetition rate electron source was generated by tightly focusing kHz, few-mJ laser pulses into an underdense plasma. This high intensity laser-plasma interaction led to stable electron beams over several hours but with strikingly complex transverse distributions even for good quality laser focal spots. Analysis of the experimental data, along with results of PIC simulations demonstrate the role of the laser wavefront on the acceleration of electrons. Distortions of the laser wavefront cause spatial inhomogeneities in the out-of-focus laser distribution and consequently, the laser pulse drives an inhomogenous transverse wakefield whose focusing/defocusing properties affect the electron distribution. These findings explain the experimental results and suggest the possibility of controlling the electron spatial distribution in laser-plasma accelerators by tailoring the laser wavefront.