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Flexible photovoltaics with a lightweight and adaptable nature that allows for deployment on curved surfaces and in building facades have always been a goal vigorously pursued by researchers in thin-film solar cell technology. The recent strides made in improving the sunlight-to-electricity conversion efficiency of kesterite Cu$_{2}$ZnSn(S, Se)$_{4}$ (CZTSSe) suggest it to be a perfect candidate. However, making use of rare Mo foil in CZTSSe solar cells causes severe problems in thermal expansion matching, uneven grain growth, and severe problems at the back contact of the devices. Herein, a strategy utilizing single-crystal graphene to modify the back interface of flexible CZTSSe solar cells is proposed. It will be shown that the insertion of graphene at the Mo foil/CZTSSe interface provides strong physical support for the subsequent deposition of the CZTSSe absorber layer, improving the adhesion between the absorber layer and the Mo foil substrate. Additionally, the graphene passivates the rough sites on the surface of the Mo foil, enhancing the chemical homogeneity of the substrate, and resulting in a more crystalline and homogeneous CZTSSe absorber layer on the Mo foil substrate. The detrimental reaction between Mo and CZTSSe has also been eliminated. Through an analysis of the electrical properties, it is found that the introduction of graphene at the back interface promotes the formation of a quasi-ohmic contact at the back contact, decreasing the back contact barrier of the solar cell, and leading to efficient collection of charges at the back interface. This investigation demonstrates that solution-based CZTSSe photovoltaic devices could form the basis of cheap and flexible solar cells.