Let $K$ be a non archimedean local field whose residue field is of characteristic $p$. Denote by $G$ the absolute Galois group of $K$. Denote by $\mu_p$ the group of $p$-roots of unity and assume it is contained in $K$.

Assume $u : G \to \mathbb{F}_p$ is a character (and thus an element of $H^1(G, \mathbb{F}_p)$). It factors through $G^{\text{ab}}/p$ which is isomorphic, via the local reciprocity map, to $K^{\times} / \left( K^{\times} \right)^p$ and thus we can see $u$ as an homomorphism $\varphi(u) : K^{\times} / \left( K^{\times} \right)^p \to \mathbb{F}_p$.

Let $\eta \in H^1(G, \mu_p)$. Kummer theory gives us an isomorphism between $H^1(G, \mu_p)$ and $K^{\times} / \left( K^{\times} \right)^p$. Hence we can see $\eta$ as an element $a(\eta) \in K^{\times} / \left( K^{\times} \right)^p$.

There is an other isomorphism in this story, namely $inv : H^2(G, \mu_p) \to \mathbb{F}_p$ which allows to identify the cup-product $$H^1(G, \mathbb{F}_p) \times H^1(G, \mu_p ) \to H^2 (G, \mu_p)$$ to the evaluation pairing $$ Hom(K^{\times} / \left( K^{\times} \right)^p, \mathbb{F}_p) \times K^{\times} / \left( K^{\times} \right)^p \to \mathbb{F}_p. $$

If $u$ and $\eta$ are such that $u \cup \eta = 0$ (and thus $\varphi(u) \left(a (\eta) \right) = 0 \in \mathbb{F}_p$, is it possible to find and "explicit" coboundary representing $u \cup v$ ?