On a four-manifold, there is apparently a relation between the first Pontryagin class modulo 4 and the Pontryagin square of the second Stiefel-Whitney class:
$\mathfrak{P}(w_2) = p_1 \; {\rm mod} \; 4$
This fact is for instance mentioned in the comments of this question, but I have been unable to find a proof of it.
My question is: Is it true that on an 8-manifold, the analogous relation
$\mathfrak{P}(w_4) \stackrel{?}{=} p_2 \; {\rm mod} \; 4$
holds?
In your first claim, it is a bit unclear what bundle you are considering. It is false for the tangent bundle of $\mathbb{C}P^2$: $\mathfrak{P}(w_2) = c_1^2 = 9 \; {\rm mod} \; 4$, while $p_1 = 3$. The context of the question you link to is arbitrary oriented rank 3 bundles over a 4-dimensional base. Is that what you have in mind?
The Pontrjagin squares in the cohomology of $BO$ were first computed by Wu in On Pontrjagin classes. III (MR0115179). It is not so easy to get hold of a digital copy of the paper or its translation, but one of the main results is Theorem 5 of Section 4:
$$\mathfrak{P}(w_{2i+1}) = \beta_4 Sq^{2i}w_{2i+1} + \theta_2(w_1 Sq^{2i}w_{2i+1})$$ $$\mathfrak{P}(w_{2i}) = \rho_4 p_i + \beta_4(w_{2i-1}w_{2i}) + \theta_2 \left(w_1 Sq^{2i-1}w_{2i} + \sum_{j=0}^{i - 1} w_{2j}w_{4i-2j}\right)$$ where $\rho_4$ is mod 4 reduction of coefficients, $\beta_4$ is the mod 4 reduction of the Bockstein map $H^i(-;Z/2) \to H^{i+1}(-; Z)$, and $\theta_2 : H^i(-;Z/2) \to H^i(-;Z/4)$ is induced by the inclusion $Z/2 \to Z/4, x \mapsto 2x$.
As a special case, we find that for any orientable vector bundle (i.e. $w_1 = 0$) $$ \mathfrak{P}(w_2) = \rho_4 p_1 + \theta_2(w_4) . $$
It's worth pointing out the corollary that Wu deduces from the above calculation (and that Stiefel-Whitney classes are homotopy invariants by Wu's formula): the mod 4 reductions of the Pontrjagin classes of a closed smooth manifold are homotopy invariants.
Ok, a more general statement holds. It is apparently due to Wu, but I found it in this paper by Emery Thomas, Theorem C.
Let $B$ be a vector bundle over a manifold $X$, $w_i$ be its Stiefel-Whitney classes and $p_i$ its Pontryagin classes. Let $\rho_4$ be the reduction modulo 4 and $\theta_2$ be the embedding of $\mathbb{Z}_2$ into $\mathbb{Z}_4$ (as well as their induced actions on cohomology groups). Then
$\mathfrak{P}(w_{2i}) = \rho_4(p_i) + \theta_2 \left( w_1 Sq^{2i-1} w_{2i} + \sum_{j = 0}^{i-1} w_{2j} w_{4i-2j} \right)$.
There is also a formula for the Pontryagin square of odd Stiefel-Whitney classes in the paper.
