Extremely large-scale array (XL-array) has emerged as a promising technology to enhance the spectrum efficiency and spatial resolution in future wireless networks, leading to a fundamental paradigm shift from conventional far-field communications towards the new near-field communications. Different from the existing works that mostly considered simultaneous wireless information and power transfer (SWIPT) in the far field, we consider in this paper a new and practical scenario, called mixed near- and far-field SWIPT, in which energy harvesting (EH) and information decoding (ID) receivers are located in the near- and far-field regions of the XL-array base station (BS), respectively. Specifically, we formulate an optimization problem to maximize the weighted sum-power harvested at all EH receivers by jointly designing the BS beam scheduling and power allocation, under the constraints on the ID sum-rate and BS transmit power. To solve this nonconvex optimization problem, an efficient algorithm is proposed to obtain a suboptimal solution by leveraging the binary variable elimination and successive convex approximation methods. Numerical results demonstrate that our proposed joint design achieves substantial performance gain over other benchmark schemes.