Yes. Indeed all irreducible components of $V\cap A$ have positive dimension. So the map is injective, since $H_{2n-2k-1}(A\setminus (V\cap A))=0$ as is shown in the answer to the following question:

A bound on the top homology of a complement to a variety in $\mathbb C^n$

Yes. Indeed all irreducible components of $V\cap A$ have positive dimension. So the map is injective, since $H_{2n-2k-1}(A\setminus (V\cap A))=0$ as is shown in the answer to the following question:

A bound on the top homology of a complement to a variety in $\mathbb C^n$