We consider the stochastic incompressible magnetohydrodynamic equations driven by additive jump noises on either the whole space $\mathbb{R}^d$, $d=2,3$ or a smooth bounded domain $D$ in $\mathbb{R}^d$. We establish the local existence and uniqueness of a mild solution in the space $L^q(0,T;\mathbb{L}^{p\otimes}_σ(D))$ allowing for initial data with less regularity, including the marginal case $u_0\in \mathbb{L}^{d\otimes}_σ(D)$. In the two-dimensional case, we also prove the global existence of mild solutions.