Under-Display Camera (UDC) has been widely exploited to help smartphones realize full-screen displays. However, as the screen could inevitably affect the light propagation process, the images captured by the UDC system usually contain flare, haze, blur, and noise. Particularly, flare and blur in UDC images could severely deteriorate the user experience in high dynamic range (HDR) scenes. In this paper, we propose a new deep model, namely UDC-UNet, to address the UDC image restoration problem with an estimated PSF in HDR scenes. Our network consists of three parts, including a U-shape base network to utilize multi-scale information, a condition branch to perform spatially variant modulation, and a kernel branch to leverage the prior knowledge of the PSF. According to the characteristics of HDR data, we additionally design a tone mapping loss to stabilize network optimization and achieve better visual quality. Experimental results show that the proposed UDC-UNet outperforms the state-of-the-art methods in quantitative and qualitative comparisons. Our approach won second place in the UDC image restoration track of the MIPI challenge. Codes and models are available at https://github.com/J-FHu/UDCUNet.