Regularization techniques have been proved useful in an enormous variety of sparse optimization problem. In this paper, we first introduce a new formulation of regularization with hybrid $L_2$-$L_p~(0 < p <1)$ norm. Then we model a sparse optimization problem with hybrid $L_2$-$L_p$ regularization. For solving the problem, we derive its local optimality conditions and develop a hybrid $L_2$-$L_p$ algorithm. Moreover, we analyze the convergence of algorithm. Finally, we apply our model and algorithm to the image recovery and deblurring for the magnetic resonance images of brain. The numerical tests show that the effects of the recovery and deblurred images, respectively, obtained by the hybrid $L_2$-$L_p$ algorithm with the suitable suitable permeate $p$ are better than those of other three algorithms, compared with the values of the signal-to-noise ratio.