  This paper is a continuation to \cite{Gazaki2017}. For every integer $n\geq
1$, we consider the generalized Galois symbol $K(k;G_1,G_2)/n\xrightarrow{s_n}
H^2(k,G_1[n]\otimes G_2[n])$, where $k$ is a finite extension of
$\mathbb{Q}_p$, $G_1,G_2$ are semi-abelian varieties over $k$ and
$K(k;G_1,G_2)$ is the Somekawa K-group attached to $G_1, G_2$. Under some mild
assumptions, we describe the exact annihilator of the image of $s_n$ under the
Tate duality perfect pairing, $H^2(k,G_1[n]\otimes G_2[n])\times
H^0(k,Hom(G_1[n]\otimes G_2[n],\mu_n))\rightarrow\mathbb{Z}/n$. An important
special case is when both $G_1, G_2$ are abelian varieties with split
semistable reduction. In this case we prove a finiteness result, which gives an
application to zero-cycles on abelian varieties and products of curves.
