In this paper, we study the following fractional Kirchhoff type equation \begin{equation*} \begin{cases} \left(a+b\displaystyle\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right)(-\Delta)_p^su=|u|^{q-2}u\ln |u|^2+\frac{\lambda}{u^\gamma}, &\rm \mathrm{in}\ \Omega, \\ u>0, &\rm \mathrm{in}\ \Omega, \\ u=0, &\rm \mathrm{in}\ \mathbb{R}^N\backslash \Omega, \end{cases} \end{equation*} where $\Omega$ $\subset$ $\mathbb{R}^N$ is a bounded domain with Lipschitz boundary, $00, b\geq0$, $N>ps$, $2p0$ is a real parameter. By using the critical point theory for nonsmooth functionals and analytic techniques, the existence and multiplicity of positive solutions are obtained.