In this paper we consider one dimensional mean-field backward stochastic differential equations (BSDEs) under weak assumptions on the coefficient. Unlike [3], the generator of our mean-field BSDEs depends not only on the solution $(Y,Z)$ but also on the law $P_{Y}$ of $Y$. The first part of the paper is devoted to the existence and uniqueness of solutions in $L^p$, $1< p\leq2$, where the monotonicity conditions are satisfied. Next, we show that if the generator $f$ is uniformly continuous in $(\mu,y,z)$, uniformly with respect to $(t,\omega)$, and if the terminal value $\xi$ belongs to $L^{p}(\Omega,\mathcal{F},P)$ with $1
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