Infrared small-target enhancement is of great significance in fields including infrared search, remote sensing, surveillance, and early warning. In single-frame infrared images, small targets usually occupy few pixels and are characterized by weak intensity, a low signal-to-clutter ratio, and limited texture information, rendering them susceptible to background clutter, nonuniform background fluctuations, strong edges and random noise. Gray-scale morphological operation and the white top-hat transform provide an effective means for background suppression and local bright small-target enhancement. However, conventional electronic implementations involve repeated sliding-window operations, neighborhood access, local comparison and intermediate data access. In this work, we propose and numerically investigate a photonic spiking neural network based on a vertical-cavity surface-emitting laser with a saturable absorber (VCSEL-SA), which can implement gray-scale morphological operation and white top-hat transform for infrared small-target enhancement. Firstly, gray-scale image is decomposed into a set of binary threshold images. Secondly, for each threshold image, the local image window and the structuring element are unfolded according to the same spatial order and temporally encoded into two rectangular optical-pulse sequences. Then, the two encoded pulse sequences are combined with a certain delay, and injected into the VCSEL-SA to implement the morphological operation including erosion and dilation according the fired spiking number. Subsequently, the binary opening operation results are stacked to reconstruct the new gray-scale image, which is used as the background estimate. As a result, small-target enhancement can be achieved by subtracting this background estimate from the original image. The spiking dynamics of VCSEL-SA is investigated to determine the feasible pulse-encoding conditions. Under Gaussian white noise with a signal-to-noise ratio (SNR) of 10 dB and input-pulse timing jitter with a standard deviation of 80 ps, the spike-response accuracy of VCSEL-SA to injected encoding pulses remains above 97%. For a gray-scale rice image, the signal-to-clutter ratio gain (SCRG), signal-to-noise ratio gain (SNRG), and background suppression factor (BSF) after small-target enhancement processing can reach 1.77, 1.46, and 2.04, respectively. The proposed method is further evaluated on the NUDT-SIRST dataset using structuring elements with different shapes and sizes. For the tested structuring elements, the maximal SCRG, SNRG, and BSF values can reach 7.73, 2.67 and 16.34, respectively. After analyzing the statistic regularities of optimized structuring-element size under different target sizes and SCRs, the optimized structuring-element size required by the small-target enhancement can be preliminarily determined based on the local statistical characteristics of original gray-scale image. Compared with traditional small-target image-enhancement methods, this proposed method can achieve relatively good indices. These results provide a feasible photonic implementation method for infrared small-target enhancement and background suppression based on a VCSEL-SA photonic spiking neural network.