Higher order embeddings of certain blow-ups of $\mathbb{P}^2$

Let $X_n$ be the blow-up of the projective plane along $n$ general points of a smooth cubic plane curve and let $\mathcal{L}$ be the linear series of strict transforms of plane curves of degree $d$ having multiplicity at least $m_i$ at the $i$-th blown-up point. We prove that if $\mathcal{L}$ is $k$-very ample, then $\mathcal{L}$ is excellent and $\mathcal{L}\cdot (-K_n) \geq k+2$. Then we give a numerical criterion for the $k$-very ampleness of excellent classes with $\mathcal{L} \cdot (-K_n) \geq k+2$, which in many cases is a necessary and sufficient condition.