Assume you have a consumption $c$ and an asset with the payoff $x$. Cochrane states that if you add "a little bit of this asset" in your portfolio first you care about the correlation between the payoff of the asset and consumption and ONLY then you care about variance. How you can see this?
Let's assume that you slighlty change your portfolio by $\xi$ (i.e. buy 0.0001 units of asset), then the variance of your consumption (which you care about) will be:
$$ \sigma^2(c + \xi x) = \sigma^2(c) + 2\xi cov(c, x) + \xi^2var(x). $$
Now, you clealry see that if $\xi$ is very small than second term will be always higher than the third term (if $var(x)$ finite). This means ih hast higher impact - first-order impact.