This paper is concerned with the following space-time fractional stochastic nonlinear partial differential equation \begin{equation*} \left(\partial_t^β+\fracν{2}\left(-Δ\right)^{α/ 2}\right) u=I_{t}^γ\Big[ f(t,x,u)-\sum_{i=1}^{d} \frac{\partial}{\partial x_i} q_i(t,x,u)+ σ(t,x,u) F_{t,x}\Big] \end{equation*} for a random field $u(t,x):[0,\infty)\times\mathbb{R}^d \mapsto\mathbb{R}$, where $α>0, β\in(0,2), γ\ge0, ν>0, F_{t,x}$ is a Lévy space-time white noise, $I_{t}^γ$ stands for the Riemann-Liouville integral in time, and $f,q_i,σ:[0,\infty)\times\mathbb{R}^d\times\mathbb{R} \mapsto\mathbb{R}$ are measurable functions. Under suitable polynomial growth conditions, we establish the existence and uniqueness of $L^2(\mathbb{R}^d)$-valued local solutions when the Lévy white noise $F_{t,x}$ contains Gaussian noise component. Furthermore, for $p\in[1,2]$, we derive the existence and uniqueness of $L^p(\mathbb{R}^d)$-valued local solutions for the equation driven by pure jump Lévy white noise. Finally, we obtain certain stronger conditions for the existence and uniqueness of global solutions.