Is there a clear interpretation of Corr(X, X+Y) in research? Consider a case of $Corr(X,Z)$, often found to be high; where later, it was found that it holds exactly $Z = X + Y$. In effect, the previously found correlations were equal to $Corr(X, X+Y)$.
How can we interpret this correlation? Is it some type of error?

If the given information is all you have, there is not much interpretation possible. There is always a random variable $Y:=Z-X$ such that $Z=X+Y$.
You said that $cor(X, Z)$ is high, which translates to $Y$ being small. Whether $Z$ can be interpreted as $X$ plus some small error $Y$ depends on the circumstances.
