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 $x\ge 3$. The proof relies on properties of random subsets of the prime numbers.