Large Gaps between Sums of Two Squareful Numbers

Let $M(x)$ be the length of the largest subinterval of $[1,x]$ which does not contain any sums of two squareful numbers. We prove a lower bound
$$M(x)\gg \frac{\ln x}{(\ln\ln x)^2}$$
for all $x\geqslant 3$. The proof relies on properties of random subsets of the prime numbers.