Use p-values to weight correlations before combining them

Question

I have a correlation between two dimensions $A$ and $B$, each dimension has 1 variable $i$ with $j$ samples. $A_i$ and $B_i$ (Let's call it $r_{Di}$). I also have a correlation of $A_i$ with another variable $C_i$ (Let's say $r_{Ei}$). For each correlation I calculate the $p$-value. For these 2 correlations, I do the correlation between ($r_D$) and ($r_E$) for all the $i$ variables. As I have the $p$-values of each correlation of $A$ with $B$ and each correlation of $A$ with $C$, can I use these $p$-values to do a weighted correlation for the correlation between $r_D$ and $r_E$? If I can use the $p$-values for this correlation how do I combine them?

I could find a question (When combining p-values, why not just averaging?) which seems to imply that there are further uses of $p$-values. But the aim of such methods seems to combine different independent studies and not from the same study.

To calculate those $p$-values are calculated with this function in r package WGCNA, whose descriptions says "Calculates Student asymptotic p-value for given correlations":

function (cor, nSamples) {
    T = sqrt(nSamples - 2) * cor/sqrt(1 - cor^2)
    2 * pt(abs(T), nSamples - 2, lower.tail = FALSE)
}

Answer 1

No, you can't do that. The flaw is that p-values do not "indicate how much I trust a value". Rather, p-values can be considered a measure of evidence against the null value. Those aren't the same thing (although they do sound similar to people at first). The confidence interval could be considered a measure of how much your trust the (point) value of your correlation, but I would not use it in the way you are thinking.

I wouldn't bother trying to use a weighted correlation at all in your circumstance. Moreover, knowing the correlations between A and B, and between A and C, does very little to constrain the possible correlation value between B and C (see: If A and B are correlated with C, why are A and B not necessarily correlated?).