We study mean-field BSDEs with jumps and a generalized mean-field operator that can capture higher order interactions. We interpret the BSDE solution as a dynamic risk measure for a representative bank whose risk attitude is influenced by the system. This influence can come in a wide class of choices, including the average system state or average intensity of system interactions. Using Fenchel-Legendre transforms, our main result is a dual representation for the expectation of the risk measure in the convex case. In particular we exhibit its dependence on the mean-field operator.