Photonics offers a transformative approach to artificial intelligence (AI) and neuromorphic computing by enabling low-latency, high-speed, and energy-efficient computations. However, conventional photonic tensor cores face significant challenges in constructing large-scale photonic neuromorphic networks. Here, we propose a fully integrated photonic tensor core, consisting of only two thin-film lithium niobate (TFLN) modulators, a III-V laser, and a charge-integration photoreceiver. Despite its simple architecture, it is capable of implementing an entire layer of a neural network with a computational speed of 120 GOPS, while also allowing flexible adjustment of the number of inputs (fan-in) and outputs (fan-out). Our tensor core supports rapid in-situ training with a weight update speed of 60 GHz. Furthermore, it successfully classifies (supervised learning) and clusters (unsupervised learning) 112 × 112-pixel images through in-situ training. To enable in-situ training for clustering AI tasks, we offer a solution for performing multiplications between two negative numbers.