Generating elements for groups and Lie algebras of the form $F/[N,N]$

Let $F$ be a free product of groups $A_i~(i\in I)$ and a free group $G$ and its normal subgroup $N$ has trivial intersection with each factor $A_i$. Subject to these conditions we will establish necessary and sufficient conditions for an element of the group $F/[N,N]$ belongs to the subgroup generated by a given finite set of elements of $F/[N,N]$ and necessary and sufficient conditions for a given set of elements of the group $F/[N,N]$ to generate it. Similar results are proved also for Lie algebras.