Wu relation for Stiefel-Whitney classes and Steenrod square is given by the following
*cochain* equation
$$
Sq^n(x_{d-n})=u_n \smile x_{d-n} + \text{d} y_{d-1} \ \text{ mod } 2
$$
on a $d$-dimensional manifold $M^d$, where
$x_{d-n} \in Z^{d-n}(M^d;\mathbb{Z_2})$ is a cocycle and
$y_{d-1} \in C^{d-1}(M^d;\mathbb{Z_2})$ is a cochain.

Also $u_n \in Z^{n}(M^d;\mathbb{Z_2})$ is the Wu class cocycle, which can be expressed in terms of Stiefel-Whitney classes: $u_1=w_1$, $u_2=w_2+w_1^2$, $u_3=w_1w_2$, etc.

**What is the explicit expression of $y_{d-1}$ in terms of
$x_{d-n}$ and Stiefel-Whitney classes?**

== added ==

A proper answer to the quenstion may need to include a part about choosing a natural cocycle for a triangulation of $M^d$ representing a Stiefel-Whitney class on $M^d$. The triangulation of $M^d$ can contain some decorations, such as the branching structure, Kasteleyn orientations, etc. See https://arxiv.org/abs/1703.10937