Meticulous modelling and performance analysis of Low-Power Wide-Area (LPWA) networks are essential for large scale dense Internet-of-Things (IoT) deployments. As Long Range (LoRa) is currently one of the most prominent LPWA technologies, we propose in this paper a stochastic-geometry-based framework to analyse the uplink transmission performance of a multi-gateway LoRa network modelled by a Matern Cluster Process (MCP). The proposed model is first to consider all together the multi-cell topology, imperfect spreading factor (SF) orthogonality, random start times, and geometric data arrival rates. Accounting for all of these factors, we initially develop the SF-dependent collision overlap time function for any start time distribution. Then, we analyse the Laplace transforms of intra-cluster and inter-cluster interference, and formulate the uplink transmission success probability. Through simulation results, we highlight the vulnerability of each SF to interference, illustrate the impact of parameters such as the network density, and the power allocation scheme on the network performance. Uniquely, our results shed light on when it is better to activate adaptive power mechanisms, as we show that an SF-based power allocation that approximates LoRa ADR, negatively impacts nodes near the cluster head. Moreover, we show that the interfering SFs degrading the performance the most depend on the decoding threshold range and the power allocation scheme.