Under the background of carbon cap-and-trade mechanism and consumer's preference for low-carbon products, we study a supply chain composed of a manufacturer with capital constraints and a retailer, and construct a stackelberg game model dominated by the manufacturer with the goal of maximizing enterprise profits. Considering three financing modes, i.e., bank financing (BF), advance payment discount (APD) and advance payment buy-back (APB), this paper investigates the emission reduction and pricing decisions of a low-carbon supply chain under different financing modes, as well as the manufacturer's financing mode selection. The research shows that: 1) No matter which financing mode the manufacturer chooses, with the increase of emission reduction investment cost coefficient, the optimal per-unit carbon emission reduction decreases. 2) The optimal carbon emission reduction level under the advance payment mode is always higher than the BF mode. When the carbon trading price is relatively high, or the emission reduction investment cost coefficient is relatively high, the manufacturer's carbon emission reduction level is the highest under the APB mode; Otherwise, the manufacturer's carbon emission reduction level is the highest under the APD mode. 3) Compared with the BF mode, advance payment can improve the profit of the manufacturer, and the profit is the highest under the APB mode.