What does the correlation function look like in first-order transition?

The correlation length at a first-order phase transition remains finite in general. It follows that the large-distance asymptotic behavior of the (truncated) 2-point correlation function is usually still of the form $$ G(r) sim frac{1}{|r|^{(d-1)/2}}e^{-|r|/xi}. $$ (The prefactor given above is the one predicted by Ornstein-Zernike theory and generally applies, but there are exceptions.)

Since $xi$ remains finite, and since there is no parameter with which to play to make $xigg 1$, one cannot investigate what happens when $1 ll |r| ll xi$. In particular, it will not be possible to reach an asymptotic regime in which the "short" distance behavior becomes universal.